src/HOL/Probability/Lebesgue_Integration.thy
author hoelzl
Mon, 23 May 2011 19:21:05 +0200
changeset 42950 6e5c2a3c69da
parent 42067 66c8281349ec
child 42991 3fa22920bf86
permissions -rw-r--r--
move lemmas to Extended_Reals and Extended_Real_Limits
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Probability/Lebesgue_Integration.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Armin Heller, TU München
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*)
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header {*Lebesgue Integration*}
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theory Lebesgue_Integration
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de0b30e6c2d2 Support product spaces on sigma finite measures.
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imports Measure Borel_Space
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begin
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41981
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lemma extreal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::extreal)"
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    13
  unfolding indicator_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    14
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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lemma tendsto_real_max:
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    16
  fixes x y :: real
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    17
  assumes "(X ---> x) net"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    18
  assumes "(Y ---> y) net"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    19
  shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
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    20
proof -
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    21
  have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
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    22
    by (auto split: split_max simp: field_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    23
  show ?thesis
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    24
    unfolding *
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    25
    by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    26
qed
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cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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lemma (in measure_space) measure_Union:
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  assumes "finite S" "S \<subseteq> sets M" "\<And>A B. A \<in> S \<Longrightarrow> B \<in> S \<Longrightarrow> A \<noteq> B \<Longrightarrow> A \<inter> B = {}"
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    30
  shows "setsum \<mu> S = \<mu> (\<Union>S)"
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    31
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    32
  have "setsum \<mu> S = \<mu> (\<Union>i\<in>S. i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    33
    using assms by (intro measure_setsum[OF `finite S`]) (auto simp: disjoint_family_on_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    34
  also have "\<dots> = \<mu> (\<Union>S)" by (auto intro!: arg_cong[where f=\<mu>])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    35
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    36
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    37
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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lemma (in sigma_algebra) measurable_sets2[intro]:
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  assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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  and "A \<in> sets M'" "B \<in> sets M''"
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    41
  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
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parents: 41831
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    42
proof -
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hoelzl
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    43
  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    44
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    45
  then show ?thesis using assms by (auto intro: measurable_sets)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    46
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    47
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    48
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
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    49
proof
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    50
  assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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qed (auto simp: incseq_def)
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    52
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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lemma borel_measurable_real_floor:
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  "(\<lambda>x::real. real \<lfloor>x\<rfloor>) \<in> borel_measurable borel"
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    55
  unfolding borel.borel_measurable_iff_ge
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    56
proof (intro allI)
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    57
  fix a :: real
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    58
  { fix x have "a \<le> real \<lfloor>x\<rfloor> \<longleftrightarrow> real \<lceil>a\<rceil> \<le> x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    59
      using le_floor_eq[of "\<lceil>a\<rceil>" x] ceiling_le_iff[of a "\<lfloor>x\<rfloor>"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    60
      unfolding real_eq_of_int by simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    61
  then have "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} = {real \<lceil>a\<rceil>..}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    62
  then show "{w::real \<in> space borel. a \<le> real \<lfloor>w\<rfloor>} \<in> sets borel" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    63
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    64
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    65
lemma measure_preservingD2:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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  "f \<in> measure_preserving A B \<Longrightarrow> f \<in> measurable A B"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    67
  unfolding measure_preserving_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    68
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    69
lemma measure_preservingD3:
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    70
  "f \<in> measure_preserving A B \<Longrightarrow> f \<in> space A \<rightarrow> space B"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    71
  unfolding measure_preserving_def measurable_def by auto
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hoelzl
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    72
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    73
lemma measure_preservingD:
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    74
  "T \<in> measure_preserving A B \<Longrightarrow> X \<in> sets B \<Longrightarrow> measure A (T -` X \<inter> space A) = measure B X"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    75
  unfolding measure_preserving_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    76
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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    77
lemma (in sigma_algebra) borel_measurable_real_natfloor[intro, simp]:
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    78
  assumes "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
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parents: 41831
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    79
  shows "(\<lambda>x. real (natfloor (f x))) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    80
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    81
  have "\<And>x. real (natfloor (f x)) = max 0 (real \<lfloor>f x\<rfloor>)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    82
    by (auto simp: max_def natfloor_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    83
  with borel_measurable_max[OF measurable_comp[OF assms borel_measurable_real_floor] borel_measurable_const]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    84
  show ?thesis by (simp add: comp_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    85
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    86
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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    87
lemma (in measure_space) AE_not_in:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
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    88
  assumes N: "N \<in> null_sets" shows "AE x. x \<notin> N"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    89
  using N by (rule AE_I') auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
    90
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    91
lemma sums_If_finite:
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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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    92
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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hoelzl
parents: 38642
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    93
  assumes finite: "finite {r. P r}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    94
  shows "(\<lambda>r. if P r then f r else 0) sums (\<Sum>r\<in>{r. P r}. f r)" (is "?F sums _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    95
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    96
  assume "{r. P r} = {}" hence "\<And>r. \<not> P r" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    97
  thus ?thesis by (simp add: sums_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
    98
next
d5d342611edb Rewrite the Probability theory.
hoelzl
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diff changeset
    99
  assume not_empty: "{r. P r} \<noteq> {}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   100
  have "?F sums (\<Sum>r = 0..< Suc (Max {r. P r}). ?F r)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   101
    by (rule series_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   102
       (auto simp add: Max_less_iff[OF finite not_empty] less_eq_Suc_le[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   103
  also have "(\<Sum>r = 0..< Suc (Max {r. P r}). ?F r) = (\<Sum>r\<in>{r. P r}. f r)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   104
    by (subst setsum_cases)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   105
       (auto intro!: setsum_cong simp: Max_ge_iff[OF finite not_empty] less_Suc_eq_le)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   106
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   107
qed
35582
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hoelzl
parents:
diff changeset
   108
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hoelzl
parents: 38642
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   109
lemma sums_single:
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
   110
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
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
   111
  shows "(\<lambda>r. if r = i then f r else 0) sums f i"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   112
  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   113
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   114
section "Simple function"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   115
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   116
text {*
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   117
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   118
Our simple functions are not restricted to positive real numbers. Instead
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   119
they are just functions with a finite range and are measurable when singleton
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   120
sets are measurable.
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   121
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   122
*}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   123
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
   124
definition "simple_function M g \<longleftrightarrow>
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   125
    finite (g ` space M) \<and>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   126
    (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   127
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   128
lemma (in sigma_algebra) simple_functionD:
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
   129
  assumes "simple_function M g"
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
   130
  shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   131
proof -
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   132
  show "finite (g ` space M)"
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   133
    using assms unfolding simple_function_def by auto
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
   134
  have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
   135
  also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
   136
  finally show "g -` X \<inter> space M \<in> sets M" using assms
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
   137
    by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def)
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   138
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   139
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   140
lemma (in sigma_algebra) simple_function_measurable2[intro]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   141
  assumes "simple_function M f" "simple_function M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   142
  shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   143
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   144
  have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   145
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   146
  then show ?thesis using assms[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   147
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   148
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   149
lemma (in sigma_algebra) simple_function_indicator_representation:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   150
  fixes f ::"'a \<Rightarrow> extreal"
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
   151
  assumes f: "simple_function M f" and x: "x \<in> space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   152
  shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   153
  (is "?l = ?r")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   154
proof -
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
   155
  have "?r = (\<Sum>y \<in> f ` space M.
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   156
    (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   157
    by (auto intro!: setsum_cong2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   158
  also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   159
    using assms by (auto dest: simple_functionD simp: setsum_delta)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   160
  also have "... = f x" using x by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   161
  finally show ?thesis by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   162
qed
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
   163
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   164
lemma (in measure_space) simple_function_notspace:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   165
  "simple_function M (\<lambda>x. h x * indicator (- space M) x::extreal)" (is "simple_function M ?h")
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   166
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   167
  have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   168
  hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   169
  have "?h -` {0} \<inter> space M = space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   170
  thus ?thesis unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   171
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   172
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   173
lemma (in sigma_algebra) simple_function_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   174
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   175
  shows "simple_function M f \<longleftrightarrow> simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   176
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   177
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   178
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   179
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   180
  thus ?thesis unfolding simple_function_def using assms by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   181
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   182
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
   183
lemma (in sigma_algebra) simple_function_cong_algebra:
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
   184
  assumes "sets N = sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   185
  shows "simple_function M f \<longleftrightarrow> simple_function N f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   186
  unfolding simple_function_def assms ..
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   187
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   188
lemma (in sigma_algebra) borel_measurable_simple_function:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   189
  assumes "simple_function M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   190
  shows "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   191
proof (rule borel_measurableI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   192
  fix S
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   193
  let ?I = "f ` (f -` S \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   194
  have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   195
  have "finite ?I"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   196
    using assms unfolding simple_function_def
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   197
    using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   198
  hence "?U \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   199
    apply (rule finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   200
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   201
  thus "f -` S \<inter> space M \<in> sets M" unfolding * .
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   202
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   203
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   204
lemma (in sigma_algebra) simple_function_borel_measurable:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   205
  fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   206
  assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   207
  shows "simple_function M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   208
  using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   209
  by (auto intro: borel_measurable_vimage)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   210
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   211
lemma (in sigma_algebra) simple_function_eq_borel_measurable:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   212
  fixes f :: "'a \<Rightarrow> extreal"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   213
  shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   214
  using simple_function_borel_measurable[of f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   215
    borel_measurable_simple_function[of f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   216
  by (fastsimp simp: simple_function_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   217
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   218
lemma (in sigma_algebra) simple_function_const[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   219
  "simple_function M (\<lambda>x. c)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   220
  by (auto intro: finite_subset simp: simple_function_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   221
lemma (in sigma_algebra) simple_function_compose[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   222
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   223
  shows "simple_function M (g \<circ> f)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   224
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   225
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   226
  show "finite ((g \<circ> f) ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   227
    using assms unfolding simple_function_def by (auto simp: image_compose)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   228
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   229
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   230
  let ?G = "g -` {g (f x)} \<inter> (f`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   231
  have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   232
    (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   233
  show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   234
    using assms unfolding simple_function_def *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   235
    by (rule_tac finite_UN) (auto intro!: finite_UN)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   236
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   237
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   238
lemma (in sigma_algebra) simple_function_indicator[intro, simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   239
  assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   240
  shows "simple_function M (indicator A)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   241
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   242
  have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   243
    by (auto simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   244
  hence "finite ?S" by (rule finite_subset) simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   245
  moreover have "- A \<inter> space M = space M - A" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   246
  ultimately show ?thesis unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   247
    using assms by (auto simp: indicator_def_raw)
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   248
qed
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   249
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   250
lemma (in sigma_algebra) simple_function_Pair[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   251
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   252
  assumes "simple_function M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   253
  shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   254
  unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   255
proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   256
  show "finite (?p ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   257
    using assms unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   258
    by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   259
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   260
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   261
  have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   262
      (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   263
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   264
  with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   265
    using assms unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   266
qed
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   267
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   268
lemma (in sigma_algebra) simple_function_compose1:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   269
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   270
  shows "simple_function M (\<lambda>x. g (f x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   271
  using simple_function_compose[OF assms, of g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   272
  by (simp add: comp_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   273
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   274
lemma (in sigma_algebra) simple_function_compose2:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   275
  assumes "simple_function M f" and "simple_function M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   276
  shows "simple_function M (\<lambda>x. h (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   277
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
   278
  have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   279
    using assms by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   280
  thus ?thesis by (simp_all add: comp_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   281
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   282
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   283
lemmas (in sigma_algebra) simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   284
  and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   285
  and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   286
  and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   287
  and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   288
  and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   289
  and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   290
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   291
lemma (in sigma_algebra) simple_function_setsum[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   292
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   293
  shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   294
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   295
  assume "finite P" from this assms show ?thesis by induct auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   296
qed auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   297
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   298
lemma (in sigma_algebra)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   299
  fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   300
  shows simple_function_extreal[intro, simp]: "simple_function M (\<lambda>x. extreal (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   301
  by (auto intro!: simple_function_compose1[OF sf])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   302
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   303
lemma (in sigma_algebra)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   304
  fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   305
  shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   306
  by (auto intro!: simple_function_compose1[OF sf])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   307
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   308
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   309
  fixes u :: "'a \<Rightarrow> extreal"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   310
  assumes u: "u \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   311
  shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   312
             (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   313
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   314
  def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   315
  { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   316
    proof (split split_if, intro conjI impI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   317
      assume "\<not> real j \<le> u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   318
      then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   319
         by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   320
      moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   321
        by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   322
      ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   323
        unfolding real_of_nat_le_iff by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   324
    qed auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   325
  note f_upper = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   326
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   327
  have real_f:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   328
    "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   329
    unfolding f_def by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   330
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   331
  let "?g j x" = "real (f x j) / 2^j :: extreal"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   332
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   333
  proof (intro exI[of _ ?g] conjI allI ballI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   334
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   335
    have "simple_function M (\<lambda>x. real (f x i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   336
    proof (intro simple_function_borel_measurable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   337
      show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   338
        using u by (auto intro!: measurable_If simp: real_f)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   339
      have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   340
        using f_upper[of _ i] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   341
      then show "finite ((\<lambda>x. real (f x i))`space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   342
        by (rule finite_subset) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   343
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   344
    then show "simple_function M (?g i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   345
      by (auto intro: simple_function_extreal simple_function_div)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   346
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   347
    show "incseq ?g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   348
    proof (intro incseq_extreal incseq_SucI le_funI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   349
      fix x and i :: nat
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   350
      have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   351
      proof ((split split_if)+, intro conjI impI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   352
        assume "extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   353
        then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   354
          by (cases "u x") (auto intro!: le_natfloor)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   355
      next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   356
        assume "\<not> extreal (real i) \<le> u x" "extreal (real (Suc i)) \<le> u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   357
        then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   358
          by (cases "u x") auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   359
      next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   360
        assume "\<not> extreal (real i) \<le> u x" "\<not> extreal (real (Suc i)) \<le> u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   361
        have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   362
          by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   363
        also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   364
        proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   365
          assume "0 \<le> u x" then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   366
            by (intro le_mult_natfloor) (cases "u x", auto intro!: mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   367
        next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   368
          assume "\<not> 0 \<le> u x" then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   369
            by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   370
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   371
        also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   372
          by (simp add: ac_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   373
        finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   374
      qed simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   375
      then show "?g i x \<le> ?g (Suc i) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   376
        by (auto simp: field_simps)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   377
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   378
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   379
    fix x show "(SUP i. ?g i x) = max 0 (u x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   380
    proof (rule extreal_SUPI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   381
      fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   382
        by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   383
                                     mult_nonpos_nonneg mult_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   384
    next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   385
      fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   386
      have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   387
      from order_trans[OF this *] have "0 \<le> y" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   388
      show "max 0 (u x) \<le> y"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   389
      proof (cases y)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   390
        case (real r)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   391
        with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   392
        from real_arch_lt[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   393
        then have "\<exists>p. max 0 (u x) = extreal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   394
        then guess p .. note ux = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   395
        obtain m :: nat where m: "p < real m" using real_arch_lt ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   396
        have "p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   397
        proof (rule ccontr)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   398
          assume "\<not> p \<le> r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   399
          with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   400
          obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   401
          then have "r * 2^max N m < p * 2^max N m - 1" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   402
          moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   403
          have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   404
            using *[of "max N m"] m unfolding real_f using ux
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   405
            by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   406
          then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   407
            by (metis real_natfloor_gt_diff_one less_le_trans)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   408
          ultimately show False by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   409
        qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   410
        then show "max 0 (u x) \<le> y" using real ux by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   411
      qed (insert `0 \<le> y`, auto)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   412
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   413
  qed (auto simp: divide_nonneg_pos)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   414
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   415
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   416
lemma (in sigma_algebra) borel_measurable_implies_simple_function_sequence':
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   417
  fixes u :: "'a \<Rightarrow> extreal"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   418
  assumes u: "u \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   419
  obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   420
    "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   421
  using borel_measurable_implies_simple_function_sequence[OF u] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   422
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   423
lemma (in sigma_algebra) simple_function_If_set:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   424
  assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   425
  shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   426
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   427
  def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   428
  show ?thesis unfolding simple_function_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   429
  proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   430
    have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   431
    from finite_subset[OF this] assms
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   432
    show "finite (?IF ` space M)" unfolding simple_function_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   433
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   434
    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
   435
    then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   436
      then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   437
      else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   438
      using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   439
    have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   440
      unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   441
    show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   442
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   443
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   444
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   445
lemma (in sigma_algebra) simple_function_If:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   446
  assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   447
  shows "simple_function M (\<lambda>x. if P x then f x else g x)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   448
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   449
  have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   450
  with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   451
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   452
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   453
lemma (in measure_space) simple_function_restricted:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   454
  fixes f :: "'a \<Rightarrow> extreal" assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   455
  shows "simple_function (restricted_space A) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator A 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
   456
    (is "simple_function ?R f \<longleftrightarrow> simple_function M ?f")
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   457
proof -
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   458
  interpret R: sigma_algebra ?R by (rule restricted_sigma_algebra[OF `A \<in> sets M`])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   459
  have f: "finite (f`A) \<longleftrightarrow> finite (?f`space M)"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   460
  proof cases
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   461
    assume "A = space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   462
    then have "f`A = ?f`space M" by (fastsimp simp: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   463
    then show ?thesis by simp
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   464
  next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   465
    assume "A \<noteq> space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   466
    then obtain x where x: "x \<in> space M" "x \<notin> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   467
      using sets_into_space `A \<in> sets M` by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   468
    have *: "?f`space M = f`A \<union> {0}"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   469
    proof (auto simp add: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   470
      show "\<exists>x\<in>space M. f x = 0 \<or> indicator A x = 0"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   471
        using x by (auto intro!: bexI[of _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   472
    next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   473
      fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   474
      then show "\<exists>y\<in>space M. f x = f y * indicator A y"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   475
        using `A \<in> sets M` sets_into_space by (auto intro!: bexI[of _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   476
    next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   477
      fix x
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   478
      assume "indicator A x \<noteq> (0::extreal)"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   479
      then have "x \<in> A" by (auto simp: indicator_def split: split_if_asm)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   480
      moreover assume "x \<in> space M" "\<forall>y\<in>A. ?f x \<noteq> f y"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   481
      ultimately show "f x = 0" by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   482
    qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   483
    then show ?thesis by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   484
  qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   485
  then show ?thesis
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   486
    unfolding simple_function_eq_borel_measurable
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   487
      R.simple_function_eq_borel_measurable
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   488
    unfolding borel_measurable_restricted[OF `A \<in> sets M`]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   489
    using assms(1)[THEN sets_into_space]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   490
    by (auto simp: indicator_def)
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   491
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   492
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   493
lemma (in sigma_algebra) simple_function_subalgebra:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   494
  assumes "simple_function N f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   495
  and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   496
  shows "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   497
  using assms unfolding simple_function_def by auto
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   498
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   499
lemma (in measure_space) simple_function_vimage:
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   500
  assumes T: "sigma_algebra M'" "T \<in> measurable M M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   501
    and f: "simple_function M' f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   502
  shows "simple_function M (\<lambda>x. f (T x))"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   503
proof (intro simple_function_def[THEN iffD2] conjI ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   504
  interpret T: sigma_algebra M' by fact
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   505
  have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   506
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   507
  then show "finite ((\<lambda>x. f (T x)) ` space M)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   508
    using f unfolding simple_function_def by (auto intro: finite_subset)
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   509
  fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   510
  then have "i \<in> f ` space M'"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   511
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   512
  then have "f -` {i} \<inter> space M' \<in> sets M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   513
    using f unfolding simple_function_def by auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   514
  then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   515
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   516
  also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   517
    using T unfolding measurable_def by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   518
  finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   519
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   520
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   521
section "Simple integral"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   522
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
   523
definition simple_integral_def:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   524
  "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * measure M (f -` {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
   525
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
   526
syntax
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   527
  "_simple_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   528
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
   529
translations
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
   530
  "\<integral>\<^isup>S x. f \<partial>M" == "CONST integral\<^isup>S M (%x. f)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   531
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   532
lemma (in measure_space) simple_integral_cong:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   533
  assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   534
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   535
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   536
  have "f ` space M = g ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   537
    "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   538
    using assms by (auto intro!: image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   539
  thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   540
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   541
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   542
lemma (in measure_space) simple_integral_cong_measure:
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
   543
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   544
    and "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   545
  shows "integral\<^isup>S N f = integral\<^isup>S M f"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   546
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
   547
  interpret v: 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
   548
    by (rule measure_space_cong) fact+
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
   549
  from simple_functionD[OF `simple_function M f`] assms 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
   550
    by (auto intro!: setsum_cong simp: simple_integral_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   551
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   552
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   553
lemma (in measure_space) simple_integral_const[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
   554
  "(\<integral>\<^isup>Sx. c \<partial>M) = c * \<mu> (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   555
proof (cases "space M = {}")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   556
  case True thus ?thesis unfolding simple_integral_def by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   557
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   558
  case False hence "(\<lambda>x. c) ` space M = {c}" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   559
  thus ?thesis unfolding simple_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   560
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   561
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   562
lemma (in measure_space) simple_function_partition:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   563
  assumes f: "simple_function M f" and g: "simple_function M g"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   564
  shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * \<mu> A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   565
    (is "_ = setsum _ (?p ` space M)")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   566
proof-
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   567
  let "?sub x" = "?p ` (f -` {x} \<inter> space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   568
  let ?SIGMA = "Sigma (f`space M) ?sub"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   569
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   570
  have [intro]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   571
    "finite (f ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   572
    "finite (g ` space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   573
    using assms unfolding simple_function_def by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   574
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   575
  { fix A
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   576
    have "?p ` (A \<inter> space M) \<subseteq>
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   577
      (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   578
      by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   579
    hence "finite (?p ` (A \<inter> space M))"
40786
0a54cfc9add3 gave more standard finite set rules simp and intro attribute
nipkow
parents: 39910
diff changeset
   580
      by (rule finite_subset) auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   581
  note this[intro, simp]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   582
  note sets = simple_function_measurable2[OF f g]
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   583
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   584
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   585
    have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   586
    with sets have "\<mu> (f -` {f x} \<inter> space M) = setsum \<mu> (?sub (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   587
      by (subst measure_Union) 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
   588
  hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * \<mu> A)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   589
    unfolding simple_integral_def using f sets
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   590
    by (subst setsum_Sigma[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   591
       (auto intro!: setsum_cong setsum_extreal_right_distrib)
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   592
  also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * \<mu> A)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   593
  proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   594
    have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   595
    have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   596
      = (\<lambda>x. (f x, ?p x)) ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   597
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   598
      fix x assume "x \<in> space M"
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   599
      thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   600
        by (auto intro!: image_eqI[of _ _ "?p x"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   601
    qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   602
    thus ?thesis
39910
10097e0a9dbd constant `contents` renamed to `the_elem`
haftmann
parents: 39302
diff changeset
   603
      apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   604
      apply (rule_tac x="xa" in image_eqI)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   605
      by simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   606
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   607
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   608
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   609
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   610
lemma (in measure_space) simple_integral_add[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   611
  assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
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
   612
  shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   613
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   614
  { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   615
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   616
    hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   617
        "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   618
      by auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   619
  with assms show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   620
    unfolding
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   621
      simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   622
      simple_function_partition[OF f g]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   623
      simple_function_partition[OF g f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   624
    by (subst (3) Int_commute)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   625
       (auto simp add: extreal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   626
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   627
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   628
lemma (in measure_space) simple_integral_setsum[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   629
  assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   630
  assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   631
  shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   632
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   633
  assume "finite P"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   634
  from this assms show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   635
    by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   636
qed auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   637
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   638
lemma (in measure_space) simple_integral_mult[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   639
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f 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
   640
  shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   641
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   642
  note mult = simple_function_mult[OF simple_function_const[of c] f(1)]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   643
  { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   644
    assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   645
    hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   646
      by auto }
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   647
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   648
    unfolding simple_function_partition[OF mult f(1)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   649
              simple_function_partition[OF f(1) mult]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   650
    by (subst setsum_extreal_right_distrib)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   651
       (auto intro!: extreal_0_le_mult setsum_cong simp: mult_assoc)
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   652
qed
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   653
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   654
lemma (in measure_space) simple_integral_mono_AE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   655
  assumes f: "simple_function M f" and g: "simple_function M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   656
  and mono: "AE x. f x \<le> g 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
   657
  shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   658
proof -
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   659
  let "?S x" = "(g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   660
  have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   661
    "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   662
  show ?thesis
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   663
    unfolding *
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   664
      simple_function_partition[OF f g]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   665
      simple_function_partition[OF g f]
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   666
  proof (safe intro!: setsum_mono)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   667
    fix x assume "x \<in> space M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   668
    then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   669
    show "the_elem (f`?S x) * \<mu> (?S x) \<le> the_elem (g`?S x) * \<mu> (?S x)"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   670
    proof (cases "f x \<le> g x")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   671
      case True then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   672
        using * assms(1,2)[THEN simple_functionD(2)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   673
        by (auto intro!: extreal_mult_right_mono)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   674
    next
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   675
      case False
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   676
      obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "\<mu> N = 0"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   677
        using mono by (auto elim!: AE_E)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   678
      have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
40871
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   679
      moreover have "?S x \<in> sets M" using assms
688f6ff859e1 Generalized simple_functionD and less_SUP_iff.
hoelzl
parents: 40859
diff changeset
   680
        by (rule_tac Int) (auto intro!: simple_functionD)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   681
      ultimately have "\<mu> (?S x) \<le> \<mu> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   682
        using `N \<in> sets M` by (auto intro!: measure_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   683
      moreover have "0 \<le> \<mu> (?S x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   684
        using assms(1,2)[THEN simple_functionD(2)] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   685
      ultimately have "\<mu> (?S x) = 0" using `\<mu> N = 0` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   686
      then show ?thesis by simp
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   687
    qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   688
  qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   689
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   690
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   691
lemma (in measure_space) simple_integral_mono:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   692
  assumes "simple_function M f" and "simple_function M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   693
  and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g 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
   694
  shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   695
  using assms by (intro simple_integral_mono_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   696
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   697
lemma (in measure_space) simple_integral_cong_AE:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   698
  assumes "simple_function M f" and "simple_function M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   699
  and "AE x. f x = g 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
   700
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   701
  using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   702
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   703
lemma (in measure_space) simple_integral_cong':
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   704
  assumes sf: "simple_function M f" "simple_function M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   705
  and mea: "\<mu> {x\<in>space M. f x \<noteq> g x} = 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
   706
  shows "integral\<^isup>S M f = integral\<^isup>S M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   707
proof (intro simple_integral_cong_AE sf AE_I)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   708
  show "\<mu> {x\<in>space M. f x \<noteq> g x} = 0" by fact
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   709
  show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   710
    using sf[THEN borel_measurable_simple_function] by auto
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   711
qed simp
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   712
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   713
lemma (in measure_space) simple_integral_indicator:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   714
  assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   715
  assumes "simple_function M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   716
  shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   717
    (\<Sum>x \<in> f ` space M. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   718
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   719
  assume "A = 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
   720
  moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   721
    by (auto intro!: simple_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   722
  moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   723
  ultimately show ?thesis by (simp add: simple_integral_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   724
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   725
  assume "A \<noteq> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   726
  then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   727
  have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   728
  proof safe
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   729
    fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   730
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   731
    fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   732
      using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   733
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   734
    show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   735
  qed
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
   736
  have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   737
    (\<Sum>x \<in> f ` space M \<union> {0}. x * \<mu> (f -` {x} \<inter> space M \<inter> A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   738
    unfolding simple_integral_def I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   739
  proof (rule setsum_mono_zero_cong_left)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   740
    show "finite (f ` space M \<union> {0})"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   741
      using assms(2) unfolding simple_function_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   742
    show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   743
      using sets_into_space[OF assms(1)] by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   744
    have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   745
      by (auto simp: image_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   746
    thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   747
      i * \<mu> (f -` {i} \<inter> space M \<inter> A) = 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   748
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   749
    fix x assume "x \<in> f`A \<union> {0}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   750
    hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   751
      by (auto simp: indicator_def split: split_if_asm)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   752
    thus "x * \<mu> (?I -` {x} \<inter> space M) =
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   753
      x * \<mu> (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   754
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   755
  show ?thesis unfolding *
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   756
    using assms(2) unfolding simple_function_def
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   757
    by (auto intro!: setsum_mono_zero_cong_right)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   758
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   759
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   760
lemma (in measure_space) simple_integral_indicator_only[simp]:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   761
  assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   762
  shows "integral\<^isup>S M (indicator A) = \<mu> A"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   763
proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   764
  assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   765
  thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   766
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   767
  assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::extreal}" by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   768
  thus ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   769
    using simple_integral_indicator[OF assms simple_function_const[of 1]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   770
    using sets_into_space[OF assms]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   771
    by (auto intro!: arg_cong[where f="\<mu>"])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   772
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   773
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   774
lemma (in measure_space) simple_integral_null_set:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   775
  assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets"
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
   776
  shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   777
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   778
  have "AE x. indicator N x = (0 :: extreal)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   779
    using `N \<in> null_sets` by (auto simp: indicator_def intro!: AE_I[of _ N])
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
   780
  then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   781
    using assms apply (intro simple_integral_cong_AE) by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   782
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   783
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   784
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   785
lemma (in measure_space) simple_integral_cong_AE_mult_indicator:
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
   786
  assumes sf: "simple_function M f" and eq: "AE x. x \<in> S" and "S \<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
   787
  shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
   788
  using assms by (intro simple_integral_cong_AE) auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
   789
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   790
lemma (in measure_space) simple_integral_restricted:
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   791
  assumes "A \<in> sets M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   792
  assumes sf: "simple_function M (\<lambda>x. f x * indicator A 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
   793
  shows "integral\<^isup>S (restricted_space A) f = (\<integral>\<^isup>Sx. f x * indicator A 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
   794
    (is "_ = integral\<^isup>S M ?f")
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   795
  unfolding simple_integral_def
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   796
proof (simp, safe intro!: setsum_mono_zero_cong_left)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   797
  from sf show "finite (?f ` space M)"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   798
    unfolding simple_function_def by auto
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   799
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   800
  fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   801
  then show "f x \<in> ?f ` space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   802
    using sets_into_space `A \<in> sets M` by (auto intro!: image_eqI[of _ _ x])
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   803
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   804
  fix x assume "x \<in> space M" "?f x \<notin> f`A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   805
  then have "x \<notin> A" by (auto simp: image_iff)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   806
  then show "?f x * \<mu> (?f -` {?f x} \<inter> space M) = 0" by simp
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   807
next
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   808
  fix x assume "x \<in> A"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   809
  then have "f x \<noteq> 0 \<Longrightarrow>
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   810
    f -` {f x} \<inter> A = ?f -` {f x} \<inter> space M"
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   811
    using `A \<in> sets M` sets_into_space
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   812
    by (auto simp: indicator_def split: split_if_asm)
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   813
  then show "f x * \<mu> (f -` {f x} \<inter> A) =
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   814
    f x * \<mu> (?f -` {f x} \<inter> space M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   815
    unfolding extreal_mult_cancel_left by auto
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   816
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   817
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
   818
lemma (in measure_space) simple_integral_subalgebra:
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
   819
  assumes N: "measure_space N" and [simp]: "space N = space M" "measure N = measure 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
   820
  shows "integral\<^isup>S N = integral\<^isup>S 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
   821
  unfolding simple_integral_def_raw by simp
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
   822
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   823
lemma (in measure_space) simple_integral_vimage:
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   824
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   825
    and f: "simple_function M' f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   826
  shows "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T x) \<partial>M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   827
proof -
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   828
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   829
  show "integral\<^isup>S M' f = (\<integral>\<^isup>S x. f (T 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
   830
    unfolding simple_integral_def
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   831
  proof (intro setsum_mono_zero_cong_right ballI)
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   832
    show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   833
      using T unfolding measurable_def measure_preserving_def by auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   834
    show "finite (f ` space M')"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   835
      using f unfolding simple_function_def by auto
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   836
  next
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   837
    fix i assume "i \<in> f ` space M' - (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   838
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = {}" by (auto simp: image_iff)
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   839
    with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   840
    show "i * T.\<mu> (f -` {i} \<inter> space M') = 0" by simp
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   841
  next
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   842
    fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M"
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   843
    then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   844
      using T unfolding measurable_def measure_preserving_def by auto
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
   845
    with f[THEN T.simple_functionD(2), THEN measure_preservingD[OF T(2)], of "{i}"]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   846
    show "i * T.\<mu> (f -` {i} \<inter> space M') = i * \<mu> ((\<lambda>x. f (T x)) -` {i} \<inter> space M)"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   847
      by auto
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
   848
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   849
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   850
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   851
lemma (in measure_space) simple_integral_cmult_indicator:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   852
  assumes A: "A \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   853
  shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * \<mu> A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   854
  using simple_integral_mult[OF simple_function_indicator[OF A]]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   855
  unfolding simple_integral_indicator_only[OF A] by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   856
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   857
lemma (in measure_space) simple_integral_positive:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   858
  assumes f: "simple_function M f" and ae: "AE x. 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   859
  shows "0 \<le> integral\<^isup>S M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   860
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   861
  have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   862
    using simple_integral_mono_AE[OF _ f ae] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   863
  then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   864
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   865
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
   866
section "Continuous positive integration"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   867
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
   868
definition positive_integral_def:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   869
  "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
   870
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
   871
syntax
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   872
  "_positive_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> extreal) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> extreal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   873
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
   874
translations
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
   875
  "\<integral>\<^isup>+ x. f \<partial>M" == "CONST integral\<^isup>P M (%x. f)"
40872
7c556a9240de Move SUP_commute, SUP_less_iff to HOL image;
hoelzl
parents: 40871
diff changeset
   876
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   877
lemma (in measure_space) positive_integral_cong_measure:
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
   878
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
   879
  shows "integral\<^isup>P N f = integral\<^isup>P M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   880
  unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   881
  unfolding simple_function_cong_algebra[OF assms(2,3), symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   882
  using AE_cong_measure[OF assms]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   883
  using simple_integral_cong_measure[OF assms]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   884
  by (auto intro!: SUP_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   885
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   886
lemma (in measure_space) positive_integral_positive:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   887
  "0 \<le> integral\<^isup>P M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   888
  by (auto intro!: le_SUPI2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   889
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   890
lemma (in measure_space) positive_integral_def_finite:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   891
  "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   892
    (is "_ = SUPR ?A ?f")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   893
  unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   894
proof (safe intro!: antisym SUP_leI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   895
  fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   896
  let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   897
  note gM = g(1)[THEN borel_measurable_simple_function]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   898
  have \<mu>G_pos: "0 \<le> \<mu> ?G" using gM by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   899
  let "?g y x" = "if g x = \<infinity> then y else max 0 (g x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   900
  from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   901
    apply (safe intro!: simple_function_max simple_function_If)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   902
    apply (force simp: max_def le_fun_def split: split_if_asm)+
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   903
    done
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   904
  show "integral\<^isup>S M g \<le> SUPR ?A ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   905
  proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   906
    have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   907
    assume "\<mu> ?G = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   908
    with gM have "AE x. x \<notin> ?G" by (simp add: AE_iff_null_set)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   909
    with gM g show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   910
      by (intro le_SUPI2[OF g0] simple_integral_mono_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   911
         (auto simp: max_def intro!: simple_function_If)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   912
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   913
    assume \<mu>G: "\<mu> ?G \<noteq> 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   914
    have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   915
    proof (intro SUP_PInfty)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   916
      fix n :: nat
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   917
      let ?y = "extreal (real n) / (if \<mu> ?G = \<infinity> then 1 else \<mu> ?G)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   918
      have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: extreal_divide_eq)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   919
      then have "?g ?y \<in> ?A" by (rule g_in_A)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   920
      have "real n \<le> ?y * \<mu> ?G"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   921
        using \<mu>G \<mu>G_pos by (cases "\<mu> ?G") (auto simp: field_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   922
      also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   923
        using `0 \<le> ?y` `?g ?y \<in> ?A` gM
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   924
        by (subst simple_integral_cmult_indicator) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   925
      also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   926
        by (intro simple_integral_mono) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   927
      finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   928
        using `?g ?y \<in> ?A` by blast
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   929
    qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   930
    then show ?thesis by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   931
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   932
qed (auto intro: le_SUPI)
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   933
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   934
lemma (in measure_space) positive_integral_mono_AE:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   935
  assumes ae: "AE x. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   936
  unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   937
proof (safe intro!: SUP_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   938
  fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   939
  from ae[THEN AE_E] guess N . note N = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   940
  then have ae_N: "AE x. x \<notin> N" by (auto intro: AE_not_in)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   941
  let "?n x" = "n x * indicator (space M - N) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   942
  have "AE x. n x \<le> ?n x" "simple_function M ?n"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   943
    using n N ae_N by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   944
  moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   945
  { fix x have "?n x \<le> max 0 (v x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   946
    proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   947
      assume x: "x \<in> space M - N"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   948
      with N have "u x \<le> v x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   949
      with n(2)[THEN le_funD, of x] x show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   950
        by (auto simp: max_def split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   951
    qed simp }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   952
  then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   953
  moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   954
    using ae_N N n by (auto intro!: simple_integral_mono_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   955
  ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   956
    by force
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   957
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   958
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   959
lemma (in measure_space) positive_integral_mono:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   960
  "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   961
  by (auto intro: positive_integral_mono_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   962
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   963
lemma (in measure_space) positive_integral_cong_AE:
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
   964
  "AE x. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   965
  by (auto simp: eq_iff intro!: positive_integral_mono_AE)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   966
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   967
lemma (in measure_space) positive_integral_cong:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   968
  "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   969
  by (auto intro: positive_integral_cong_AE)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
   970
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   971
lemma (in measure_space) positive_integral_eq_simple_integral:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   972
  assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   973
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   974
  let "?f x" = "f x * indicator (space M) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   975
  have f': "simple_function M ?f" using f by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   976
  with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   977
    by (auto simp: fun_eq_iff max_def split: split_indicator)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   978
  have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   979
    by (force intro!: SUP_leI simple_integral_mono simp: le_fun_def positive_integral_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   980
  moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   981
    unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   982
    using f' by (auto intro!: le_SUPI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   983
  ultimately show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   984
    by (simp cong: positive_integral_cong simple_integral_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   985
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   986
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   987
lemma (in measure_space) positive_integral_eq_simple_integral_AE:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   988
  assumes f: "simple_function M f" "AE x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   989
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   990
  have "AE x. f x = max 0 (f x)" using f by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   991
  with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   992
    by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   993
             add: positive_integral_eq_simple_integral)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   994
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   995
    by (auto intro!: simple_integral_cong_AE split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   996
qed
40873
1ef85f4e7097 Shorter definition for positive_integral.
hoelzl
parents: 40872
diff changeset
   997
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
   998
lemma (in measure_space) positive_integral_SUP_approx:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
   999
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1000
  and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1001
  shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1002
proof (rule extreal_le_mult_one_interval)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1003
  have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1004
    using f(3) by (auto intro!: le_SUPI2 positive_integral_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1005
  then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1006
  have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1007
    using u(3) by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1008
  fix a :: extreal assume "0 < a" "a < 1"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1009
  hence "a \<noteq> 0" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1010
  let "?B i" = "{x \<in> space M. a * u x \<le> f i x}"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1011
  have B: "\<And>i. ?B i \<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
  1012
    using f `simple_function M u` by (auto simp: borel_measurable_simple_function)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1013
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1014
  let "?uB i x" = "u x * indicator (?B i) x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1015
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1016
  { fix i have "?B i \<subseteq> ?B (Suc i)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1017
    proof safe
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1018
      fix i x assume "a * u x \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1019
      also have "\<dots> \<le> f (Suc i) x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1020
        using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1021
      finally show "a * u x \<le> f (Suc i) x" .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1022
    qed }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1023
  note B_mono = this
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1024
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1025
  note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1026
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1027
  let "?B' i n" = "(u -` {i} \<inter> space M) \<inter> ?B n"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1028
  have measure_conv: "\<And>i. \<mu> (u -` {i} \<inter> space M) = (SUP n. \<mu> (?B' i n))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1029
  proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1030
    fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1031
    have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1032
    have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1033
    have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1034
    proof safe
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1035
      fix x i assume x: "x \<in> space M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1036
      show "x \<in> (\<Union>i. ?B' (u x) i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1037
      proof cases
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1038
        assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1039
      next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1040
        assume "u x \<noteq> 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1041
        with `a < 1` u_range[OF `x \<in> space M`]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1042
        have "a * u x < 1 * u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1043
          by (intro extreal_mult_strict_right_mono) (auto simp: image_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1044
        also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def SUPR_apply)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1045
        finally obtain i where "a * u x < f i x" unfolding SUPR_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1046
          by (auto simp add: less_Sup_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1047
        hence "a * u x \<le> f i x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1048
        thus ?thesis using `x \<in> space M` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1049
      qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1050
    qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1051
    then show "?thesis i" using continuity_from_below[OF 1 2] by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1052
  qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1053
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
  1054
  have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1055
    unfolding simple_integral_indicator[OF B `simple_function M u`]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1056
  proof (subst SUPR_extreal_setsum, safe)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1057
    fix x n 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
    with u_range show "incseq (\<lambda>i. u x * \<mu> (?B' (u x) i))" "\<And>i. 0 \<le> u x * \<mu> (?B' (u x) i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1059
      using B_mono B_u by (auto intro!: measure_mono extreal_mult_left_mono incseq_SucI simp: extreal_zero_le_0_iff)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1060
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1061
    show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * \<mu> (?B' i n))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1062
      using measure_conv u_range B_u unfolding simple_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1063
      by (auto intro!: setsum_cong SUPR_extreal_cmult[symmetric])
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1064
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1065
  moreover
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
  1066
  have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1067
    apply (subst SUPR_extreal_cmult[symmetric])
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  1068
  proof (safe intro!: SUP_mono bexI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1069
    fix i
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1070
    have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1071
      using B `simple_function M u` u_range
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1072
      by (subst simple_integral_mult) (auto split: split_indicator)
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
  1073
    also have "\<dots> \<le> integral\<^isup>P M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1074
    proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1075
      have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1076
      show ?thesis using f(3) * u_range `0 < a`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1077
        by (subst positive_integral_eq_simple_integral[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1078
           (auto intro!: positive_integral_mono split: split_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1079
    qed
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
  1080
    finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1081
      by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1082
  next
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1083
    fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1084
      by (intro simple_integral_positive) (auto split: split_indicator)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1085
  qed (insert `0 < a`, 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
  1086
  ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1087
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1088
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1089
lemma (in measure_space) incseq_positive_integral:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1090
  assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1091
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1092
  have "\<And>i x. f i x \<le> f (Suc i) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1093
    using assms by (auto dest!: incseq_SucD simp: le_fun_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1094
  then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1095
    by (auto intro!: incseq_SucI positive_integral_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1096
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1097
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1098
text {* Beppo-Levi monotone convergence theorem *}
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1099
lemma (in measure_space) positive_integral_monotone_convergence_SUP:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1100
  assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1101
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1102
proof (rule antisym)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1103
  show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1104
    by (auto intro!: SUP_leI le_SUPI positive_integral_mono)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1105
next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1106
  show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1107
    unfolding positive_integral_def_finite[of "\<lambda>x. SUP i. f i x"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1108
  proof (safe intro!: SUP_leI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1109
    fix g assume g: "simple_function M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1110
      and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1111
    moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1112
      using f by (auto intro!: le_SUPI2)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1113
    ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1114
      by (intro  positive_integral_SUP_approx[OF f g _ g'])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1115
         (auto simp: le_fun_def max_def SUPR_apply)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1116
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1117
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1118
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1119
lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1120
  assumes f: "\<And>i. AE x. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1121
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1122
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1123
  from f have "AE x. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1124
    by (simp add: AE_all_countable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1125
  from this[THEN AE_E] guess N . note N = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1126
  let "?f i x" = "if x \<in> space M - N then f i x else 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1127
  have f_eq: "AE x. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ N])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1128
  then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1129
    by (auto intro!: positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1130
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1131
  proof (rule positive_integral_monotone_convergence_SUP)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1132
    show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1133
    { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1134
        using f N(3) by (intro measurable_If_set) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1135
      fix x show "0 \<le> ?f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1136
        using N(1) by auto }
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1137
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1138
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1139
    using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1140
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1141
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1142
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1143
lemma (in measure_space) positive_integral_monotone_convergence_SUP_AE_incseq:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1144
  assumes f: "incseq f" "\<And>i. AE x. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1145
  shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1146
  using f[unfolded incseq_Suc_iff le_fun_def]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1147
  by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1148
     auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1149
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1150
lemma (in measure_space) positive_integral_monotone_convergence_simple:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1151
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1152
  shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1153
  using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1154
    f(3)[THEN borel_measurable_simple_function] f(2)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1155
  by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1156
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1157
lemma positive_integral_max_0:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1158
  "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1159
  by (simp add: le_fun_def positive_integral_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1160
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1161
lemma (in measure_space) positive_integral_cong_pos:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1162
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1163
  shows "integral\<^isup>P M f = integral\<^isup>P M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1164
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1165
  have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1166
  proof (intro positive_integral_cong)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1167
    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
  1168
    from assms[OF this] show "max 0 (f x) = max 0 (g x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1169
      by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1170
  qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1171
  then show ?thesis by (simp add: positive_integral_max_0)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1172
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1173
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1174
lemma (in measure_space) SUP_simple_integral_sequences:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1175
  assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1176
  and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1177
  and eq: "AE x. (SUP i. f i x) = (SUP i. g i x)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1178
  shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1179
    (is "SUPR _ ?F = SUPR _ ?G")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1180
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1181
  have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1182
    using f by (rule positive_integral_monotone_convergence_simple)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1183
  also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1184
    unfolding eq[THEN positive_integral_cong_AE] ..
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1185
  also have "\<dots> = (SUP i. ?G i)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1186
    using g by (rule positive_integral_monotone_convergence_simple[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1187
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1188
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1189
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1190
lemma (in measure_space) positive_integral_const[simp]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1191
  "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * \<mu> (space M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1192
  by (subst positive_integral_eq_simple_integral) auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1193
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1194
lemma (in measure_space) positive_integral_vimage:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1195
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1196
  and f: "f \<in> borel_measurable M'"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1197
  shows "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1198
proof -
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1199
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1200
  from T.borel_measurable_implies_simple_function_sequence'[OF f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1201
  guess f' . note f' = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1202
  let "?f i x" = "f' i (T x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1203
  have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1204
  have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1205
    using f'(4) .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1206
  have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1207
    using simple_function_vimage[OF T(1) measure_preservingD2[OF T(2)] f'(1)] .
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
  1208
  show "integral\<^isup>P M' f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1209
    using
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1210
      T.positive_integral_monotone_convergence_simple[OF f'(2,5,1)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1211
      positive_integral_monotone_convergence_simple[OF inc f'(5) sf]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1212
    by (simp add: positive_integral_max_0 simple_integral_vimage[OF T f'(1)] f')
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1213
qed
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1214
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1215
lemma (in measure_space) positive_integral_linear:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1216
  assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1217
  and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g 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
  1218
  shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1219
    (is "integral\<^isup>P M ?L = _")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1220
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1221
  from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1222
  note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1223
  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1224
  note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1225
  let "?L' i x" = "a * u i x + v i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1226
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1227
  have "?L \<in> borel_measurable M" using assms by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1228
  from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1229
  note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1230
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1231
  have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1232
    using u v `0 \<le> a`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1233
    by (auto simp: incseq_Suc_iff le_fun_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1234
             intro!: add_mono extreal_mult_left_mono simple_integral_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1235
  have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1236
    using u v `0 \<le> a` by (auto simp: simple_integral_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1237
  { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1238
      by (auto split: split_if_asm) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1239
  note not_MInf = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1240
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1241
  have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1242
  proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1243
    show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1244
      using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1245
      by (auto intro!: add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1246
    { fix x
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1247
      { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1248
          by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1249
      then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1250
        using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1251
        by (subst SUPR_extreal_cmult[symmetric, OF u(6) `0 \<le> a`])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1252
           (auto intro!: SUPR_extreal_add
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1253
                 simp: incseq_Suc_iff le_fun_def add_mono extreal_mult_left_mono extreal_add_nonneg_nonneg) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1254
    then show "AE x. (SUP i. l i x) = (SUP i. ?L' i x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1255
      unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1256
      by (intro AE_I2) (auto split: split_max simp add: extreal_add_nonneg_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1257
  qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1258
  also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1259
    using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1260
  finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1261
    unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1262
    unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1263
    apply (subst SUPR_extreal_cmult[symmetric, OF pos(1) `0 \<le> a`])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1264
    apply (subst SUPR_extreal_add[symmetric, OF inc not_MInf]) .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1265
  then show ?thesis by (simp add: positive_integral_max_0)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1266
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1267
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1268
lemma (in measure_space) positive_integral_cmult:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1269
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
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
  1270
  shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1271
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1272
  have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1273
    by (auto split: split_max simp: extreal_zero_le_0_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1274
  have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1275
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1276
  then show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1277
    using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" "\<lambda>x. 0"] f
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1278
    by (auto simp: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1279
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1280
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1281
lemma (in measure_space) positive_integral_multc:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1282
  assumes "f \<in> borel_measurable M" "AE x. 0 \<le> f x" "0 \<le> c"
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
  1283
  shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1284
  unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1285
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1286
lemma (in measure_space) positive_integral_indicator[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
  1287
  "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = \<mu> A"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1288
  by (subst positive_integral_eq_simple_integral)
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1289
     (auto simp: simple_function_indicator simple_integral_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1290
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1291
lemma (in measure_space) positive_integral_cmult_indicator:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1292
  "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * \<mu> A"
41544
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1293
  by (subst positive_integral_eq_simple_integral)
c3b977fee8a3 introduced integral syntax
hoelzl
parents: 41097
diff changeset
  1294
     (auto simp: simple_function_indicator simple_integral_indicator)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1295
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1296
lemma (in measure_space) positive_integral_add:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1297
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1298
  and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g 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
  1299
  shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1300
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1301
  have ae: "AE x. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1302
    using assms by (auto split: split_max simp: extreal_add_nonneg_nonneg)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1303
  have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1304
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1305
  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1306
    unfolding ae[THEN positive_integral_cong_AE] ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1307
  also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1308
    using positive_integral_linear[of "\<lambda>x. max 0 (f x)" 1 "\<lambda>x. max 0 (g x)"] f g
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1309
    by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1310
  finally show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1311
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1312
qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1313
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1314
lemma (in measure_space) positive_integral_setsum:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1315
  assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x. 0 \<le> f i x"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1316
  shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1317
proof cases
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1318
  assume f: "finite P"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1319
  from assms have "AE x. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1320
  from f this assms(1) show ?thesis
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1321
  proof induct
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1322
    case (insert i P)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1323
    then have "f i \<in> borel_measurable M" "AE x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1324
      "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x. 0 \<le> (\<Sum>i\<in>P. f i x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1325
      by (auto intro!: borel_measurable_extreal_setsum setsum_nonneg)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1326
    from positive_integral_add[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1327
    show ?case using insert by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1328
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1329
qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1330
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1331
lemma (in measure_space) positive_integral_Markov_inequality:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1332
  assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" "c \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1333
  shows "\<mu> ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1334
    (is "\<mu> ?A \<le> _ * ?PI")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1335
proof -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1336
  have "?A \<in> sets M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1337
    using `A \<in> sets M` u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1338
  hence "\<mu> ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1339
    using positive_integral_indicator by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1340
  also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1341
    by (auto intro!: positive_integral_mono_AE
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1342
      simp: indicator_def extreal_zero_le_0_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1343
  also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1344
    using assms
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1345
    by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: extreal_zero_le_0_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1346
  finally show ?thesis .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1347
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1348
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1349
lemma (in measure_space) positive_integral_noteq_infinite:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1350
  assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1351
  and "integral\<^isup>P M g \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1352
  shows "AE x. g x \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1353
proof (rule ccontr)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1354
  assume c: "\<not> (AE x. g x \<noteq> \<infinity>)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1355
  have "\<mu> {x\<in>space M. g x = \<infinity>} \<noteq> 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1356
    using c g by (simp add: AE_iff_null_set)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1357
  moreover have "0 \<le> \<mu> {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1358
  ultimately have "0 < \<mu> {x\<in>space M. g x = \<infinity>}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1359
  then have "\<infinity> = \<infinity> * \<mu> {x\<in>space M. g x = \<infinity>}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1360
  also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1361
    using g by (subst positive_integral_cmult_indicator) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1362
  also have "\<dots> \<le> integral\<^isup>P M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1363
    using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1364
  finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1365
qed
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1366
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1367
lemma (in measure_space) positive_integral_diff:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1368
  assumes f: "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1369
  and g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1370
  and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1371
  and mono: "AE x. g x \<le> f 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
  1372
  shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1373
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1374
  have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x. 0 \<le> f x - g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1375
    using assms by (auto intro: extreal_diff_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1376
  have pos_f: "AE x. 0 \<le> f x" using mono g by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1377
  { fix a b :: extreal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1378
      by (cases rule: extreal2_cases[of a b]) auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1379
  note * = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1380
  then have "AE x. f x = f x - g x + g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1381
    using mono positive_integral_noteq_infinite[OF g fin] assms by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1382
  then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1383
    unfolding positive_integral_add[OF diff g, symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1384
    by (rule positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1385
  show ?thesis unfolding **
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1386
    using fin positive_integral_positive[of g]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1387
    by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1388
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1389
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1390
lemma (in measure_space) positive_integral_suminf:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1391
  assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1392
  shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1393
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1394
  have all_pos: "AE x. \<forall>i. 0 \<le> f i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1395
    using assms by (auto simp: AE_all_countable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1396
  have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1397
    using positive_integral_positive by (rule suminf_extreal_eq_SUPR)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1398
  also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1399
    unfolding positive_integral_setsum[OF f] ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1400
  also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1401
    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1402
       (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1403
  also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1404
    by (intro positive_integral_cong_AE) (auto simp: suminf_extreal_eq_SUPR)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1405
  finally show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1406
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1407
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1408
text {* Fatou's lemma: convergence theorem on limes inferior *}
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1409
lemma (in measure_space) positive_integral_lim_INF:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1410
  fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> extreal"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1411
  assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x. 0 \<le> u i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1412
  shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1413
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1414
  have pos: "AE x. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1415
  have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1416
    (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1417
    unfolding liminf_SUPR_INFI using pos u
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1418
    by (intro positive_integral_monotone_convergence_SUP_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1419
       (elim AE_mp, auto intro!: AE_I2 intro: le_INFI INF_subset)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1420
  also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1421
    unfolding liminf_SUPR_INFI
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1422
    by (auto intro!: SUP_mono exI le_INFI positive_integral_mono INF_leI)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1423
  finally show ?thesis .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1424
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1425
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1426
lemma (in measure_space) measure_space_density:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1427
  assumes u: "u \<in> borel_measurable M" "AE x. 0 \<le> u 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
  1428
    and M'[simp]: "M' = (M\<lparr>measure := \<lambda>A. (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)\<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
  1429
  shows "measure_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
  1430
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
  1431
  interpret M': sigma_algebra M' by (intro 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
  1432
  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
  1433
  proof
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1434
    have pos: "\<And>A. AE x. 0 \<le> u x * indicator A x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1435
      using u by (auto simp: extreal_zero_le_0_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1436
    then show "positive M' (measure M')" unfolding M'
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1437
      using u(1) by (auto simp: positive_def intro!: positive_integral_positive)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1438
    show "countably_additive M' (measure 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
  1439
    proof (intro countably_additiveI)
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
  1440
      fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M'"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1441
      then have *: "\<And>i. (\<lambda>x. u x * indicator (A i) x) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1442
        using u by (auto intro: borel_measurable_indicator)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1443
      assume disj: "disjoint_family A"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1444
      have "(\<Sum>n. measure M' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. u x * indicator (A n) x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1445
        unfolding M' using u(1) *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1446
        by (simp add: positive_integral_suminf[OF _ pos, symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1447
      also have "\<dots> = (\<integral>\<^isup>+ x. u x * (\<Sum>n. indicator (A n) x) \<partial>M)" using u
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1448
        by (intro positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1449
           (elim AE_mp, auto intro!: AE_I2 suminf_cmult_extreal)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1450
      also have "\<dots> = (\<integral>\<^isup>+ x. u x * indicator (\<Union>n. A n) x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1451
        unfolding suminf_indicator[OF disj] ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1452
      finally show "(\<Sum>n. measure M' (A n)) = measure M' (\<Union>x. A x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1453
        unfolding M' 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
  1454
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1455
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1456
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1457
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1458
lemma (in measure_space) positive_integral_null_set:
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
  1459
  assumes "N \<in> null_sets" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1460
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
  1461
  have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1462
  proof (intro positive_integral_cong_AE AE_I)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1463
    show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1464
      by (auto simp: indicator_def)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1465
    show "\<mu> N = 0" "N \<in> sets M"
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1466
      using assms by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1467
  qed
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1468
  then show ?thesis by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1469
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1470
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1471
lemma (in measure_space) positive_integral_translated_density:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1472
  assumes f: "f \<in> borel_measurable M" "AE x. 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1473
  assumes g: "g \<in> borel_measurable M" "AE x. 0 \<le> g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1474
    and M': "M' = (M\<lparr> measure := \<lambda>A. (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)\<rparr>)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1475
  shows "integral\<^isup>P M' g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1476
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1477
  from measure_space_density[OF f M']
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1478
  interpret T: measure_space M' .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1479
  have borel[simp]:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1480
    "borel_measurable M' = borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1481
    "simple_function M' = simple_function M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1482
    unfolding measurable_def simple_function_def_raw by (auto simp: M')
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1483
  from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1484
  note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1485
  note G'(2)[simp]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1486
  { fix P have "AE x. P x \<Longrightarrow> AE x in M'. P x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1487
      using positive_integral_null_set[of _ f]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1488
      unfolding T.almost_everywhere_def almost_everywhere_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1489
      by (auto simp: M') }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1490
  note ac = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1491
  from G(4) g(2) have G_M': "AE x in M'. (SUP i. G i x) = g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1492
    by (auto intro!: ac split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1493
  { fix i
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1494
    let "?I y x" = "indicator (G i -` {y} \<inter> space M) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1495
    { fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1496
      then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1497
      from * G' G have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * (\<Sum>y\<in>G i`space M. (y * ?I y x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1498
        by (subst setsum_extreal_right_distrib) (auto simp: ac_simps)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1499
      also have "\<dots> = f x * G i x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1500
        by (simp add: indicator_def if_distrib setsum_cases)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1501
      finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1502
    note to_singleton = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1503
    have "integral\<^isup>P M' (G i) = integral\<^isup>S M' (G i)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1504
      using G T.positive_integral_eq_simple_integral by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1505
    also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1506
      unfolding simple_integral_def M' by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1507
    also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1508
      using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1509
    also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1510
      using f G' G by (auto intro!: positive_integral_setsum[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1511
    finally have "integral\<^isup>P M' (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1512
      using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1513
  note [simp] = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1514
  have "integral\<^isup>P M' g = (SUP i. integral\<^isup>P M' (G i))" using G'(1) G_M'(1) G
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1515
    using T.positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1516
    by (simp cong: T.positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1517
  also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1518
  also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1519
    using f G' G(2)[THEN incseq_SucD] G
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1520
    by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1521
       (auto simp: extreal_mult_left_mono le_fun_def extreal_zero_le_0_iff)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1522
  also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1523
    by (intro positive_integral_cong_AE)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1524
       (auto simp add: SUPR_extreal_cmult split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1525
  finally show "integral\<^isup>P M' g = (\<integral>\<^isup>+x. f x * g x \<partial>M)" .
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1526
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1527
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1528
lemma (in measure_space) positive_integral_0_iff:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1529
  assumes u: "u \<in> borel_measurable M" and pos: "AE x. 0 \<le> u 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
  1530
  shows "integral\<^isup>P M u = 0 \<longleftrightarrow> \<mu> {x\<in>space M. u x \<noteq> 0} = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1531
    (is "_ \<longleftrightarrow> \<mu> ?A = 0")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1532
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1533
  have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1534
    by (auto intro!: positive_integral_cong simp: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1535
  show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1536
  proof
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1537
    assume "\<mu> ?A = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1538
    with positive_integral_null_set[of ?A u] u
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1539
    show "integral\<^isup>P M u = 0" by (simp add: u_eq)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1540
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1541
    { fix r :: extreal and n :: nat assume gt_1: "1 \<le> real n * r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1542
      then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_extreal_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1543
      then have "0 \<le> r" by (auto simp add: extreal_zero_less_0_iff) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1544
    note gt_1 = this
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1545
    assume *: "integral\<^isup>P M u = 0"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1546
    let "?M n" = "{x \<in> space M. 1 \<le> real (n::nat) * u x}"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1547
    have "0 = (SUP n. \<mu> (?M n \<inter> ?A))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1548
    proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1549
      { fix n :: nat
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1550
        from positive_integral_Markov_inequality[OF u pos, of ?A "extreal (real n)"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1551
        have "\<mu> (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1552
        moreover have "0 \<le> \<mu> (?M n \<inter> ?A)" using u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1553
        ultimately have "\<mu> (?M n \<inter> ?A) = 0" by auto }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1554
      thus ?thesis by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1555
    qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1556
    also have "\<dots> = \<mu> (\<Union>n. ?M n \<inter> ?A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1557
    proof (safe intro!: continuity_from_below)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1558
      fix n show "?M n \<inter> ?A \<in> sets M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1559
        using u by (auto intro!: Int)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1560
    next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1561
      show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1562
      proof (safe intro!: incseq_SucI)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1563
        fix n :: nat and x
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1564
        assume *: "1 \<le> real n * u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1565
        also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1566
          using `0 \<le> u x` by (auto intro!: extreal_mult_right_mono)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1567
        finally show "1 \<le> real (Suc n) * u x" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1568
      qed
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1569
    qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1570
    also have "\<dots> = \<mu> {x\<in>space M. 0 < u x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1571
    proof (safe intro!: arg_cong[where f="\<mu>"] dest!: gt_1)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1572
      fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1573
      show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1574
      proof (cases "u x")
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1575
        case (real r) with `0 < u x` have "0 < r" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1576
        obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1577
        hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1578
        hence "1 \<le> real j * r" using real `0 < r` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1579
        thus ?thesis using `0 < r` real by (auto simp: one_extreal_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1580
      qed (insert `0 < u x`, auto)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1581
    qed auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1582
    finally have "\<mu> {x\<in>space M. 0 < u x} = 0" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1583
    moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1584
    from pos have "AE x. \<not> (u x < 0)" by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1585
    then have "\<mu> {x\<in>space M. u x < 0} = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1586
      using AE_iff_null_set u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1587
    moreover have "\<mu> {x\<in>space M. u x \<noteq> 0} = \<mu> {x\<in>space M. u x < 0} + \<mu> {x\<in>space M. 0 < u x}"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1588
      using u by (subst measure_additive) (auto intro!: arg_cong[where f=\<mu>])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1589
    ultimately show "\<mu> ?A = 0" by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1590
  qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1591
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1592
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1593
lemma (in measure_space) positive_integral_0_iff_AE:
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1594
  assumes u: "u \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1595
  shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. u x \<le> 0)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1596
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1597
  have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1598
    using u by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1599
  from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1600
  have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x. max 0 (u x) = 0)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1601
    unfolding positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1602
    using AE_iff_null_set[OF sets] u by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1603
  also have "\<dots> \<longleftrightarrow> (AE x. u x \<le> 0)" by (auto split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1604
  finally show ?thesis .
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1605
qed
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1606
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1607
lemma (in measure_space) positive_integral_restricted:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1608
  assumes A: "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
  1609
  shows "integral\<^isup>P (restricted_space A) f = (\<integral>\<^isup>+ x. f x * indicator A 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
  1610
    (is "integral\<^isup>P ?R f = integral\<^isup>P M ?f")
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1611
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1612
  interpret R: measure_space ?R
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1613
    by (rule restricted_measure_space) fact
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1614
  let "?I g x" = "g x * indicator A x :: extreal"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1615
  show ?thesis
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1616
    unfolding positive_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1617
    unfolding simple_function_restricted[OF A]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1618
    unfolding AE_restricted[OF A]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1619
  proof (safe intro!: SUPR_eq)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1620
    fix g assume g: "simple_function M (?I g)" and le: "g \<le> max 0 \<circ> f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1621
    show "\<exists>j\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> ?I f}.
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1622
      integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M j"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1623
    proof (safe intro!: bexI[of _ "?I g"])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1624
      show "integral\<^isup>S (restricted_space A) g \<le> integral\<^isup>S M (?I g)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1625
        using g A by (simp add: simple_integral_restricted)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1626
      show "?I g \<le> max 0 \<circ> ?I f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1627
        using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1628
    qed fact
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1629
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1630
    fix g assume g: "simple_function M g" and le: "g \<le> max 0 \<circ> ?I f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1631
    show "\<exists>i\<in>{g. simple_function M (?I g) \<and> g \<le> max 0 \<circ> f}.
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1632
      integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) i"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1633
    proof (safe intro!: bexI[of _ "?I g"])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1634
      show "?I g \<le> max 0 \<circ> f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1635
        using le by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1636
      from le have "\<And>x. g x \<le> ?I (?I g) x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1637
        by (auto simp: le_fun_def max_def indicator_def split: split_if_asm)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1638
      then show "integral\<^isup>S M g \<le> integral\<^isup>S (restricted_space A) (?I g)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1639
        using A g by (auto intro!: simple_integral_mono simp: simple_integral_restricted)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1640
      show "simple_function M (?I (?I g))" using g A by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1641
    qed
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1642
  qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1643
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1644
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1645
lemma (in measure_space) positive_integral_subalgebra:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1646
  assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f 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
  1647
  and N: "sets N \<subseteq> sets M" "space N = space M" "\<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
  1648
  and sa: "sigma_algebra 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
  1649
  shows "integral\<^isup>P N f = integral\<^isup>P M f"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1650
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
  1651
  interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1652
  from N.borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1653
  note sf = simple_function_subalgebra[OF fs(1) N(1,2)]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1654
  from N.positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric]
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
  1655
  have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * N.\<mu> (fs i -` {x} \<inter> space M))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1656
    unfolding fs(4) positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1657
    unfolding simple_integral_def `space N = space M` 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
  1658
  also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * \<mu> (fs i -` {x} \<inter> space M))"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1659
    using N N.simple_functionD(2)[OF fs(1)] unfolding `space N = space M` by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1660
  also have "\<dots> = integral\<^isup>P M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1661
    using positive_integral_monotone_convergence_simple[OF fs(2,5) sf, symmetric]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1662
    unfolding fs(4) positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1663
    unfolding simple_integral_def `space N = space M` 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
  1664
  finally show ?thesis .
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1665
qed
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  1666
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1667
section "Lebesgue Integral"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1668
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
  1669
definition integrable where
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
  1670
  "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1671
    (\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1672
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
  1673
lemma integrableD[dest]:
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1674
  assumes "integrable M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1675
  shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. extreal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. extreal (- f x) \<partial>M) \<noteq> \<infinity>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1676
  using assms unfolding integrable_def by auto
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  1677
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
  1678
definition lebesgue_integral_def:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1679
  "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. extreal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. extreal (- f x) \<partial>M))"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1680
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
  1681
syntax
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
  1682
  "_lebesgue_integral" :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a, 'b) measure_space_scheme \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
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
  1683
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
  1684
translations
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
  1685
  "\<integral> x. f \<partial>M" == "CONST integral\<^isup>L M (%x. f)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1686
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1687
lemma (in measure_space) integrableE:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1688
  assumes "integrable M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1689
  obtains r q where
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1690
    "(\<integral>\<^isup>+x. extreal (f x)\<partial>M) = extreal r"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1691
    "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M) = extreal q"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1692
    "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1693
  using assms unfolding integrable_def lebesgue_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1694
  using positive_integral_positive[of "\<lambda>x. extreal (f x)"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1695
  using positive_integral_positive[of "\<lambda>x. extreal (-f x)"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1696
  by (cases rule: extreal2_cases[of "(\<integral>\<^isup>+x. extreal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. extreal (f x)\<partial>M)"]) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1697
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1698
lemma (in measure_space) integral_cong:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1699
  assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g 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
  1700
  shows "integral\<^isup>L M f = integral\<^isup>L M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1701
  using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1702
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1703
lemma (in measure_space) integral_cong_measure:
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
  1704
  assumes "\<And>A. A \<in> sets M \<Longrightarrow> measure N A = \<mu> A" "sets N = sets M" "space N = space M"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1705
  shows "integral\<^isup>L N f = integral\<^isup>L M f"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1706
  by (simp add: positive_integral_cong_measure[OF assms] lebesgue_integral_def)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1707
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1708
lemma (in measure_space) integral_cong_AE:
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1709
  assumes cong: "AE x. f x = g 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
  1710
  shows "integral\<^isup>L M f = integral\<^isup>L M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1711
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1712
  have *: "AE x. extreal (f x) = extreal (g x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1713
    "AE x. extreal (- f x) = extreal (- g x)" using cong by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1714
  show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1715
    unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1716
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1717
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1718
lemma (in measure_space) integrable_cong:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1719
  "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1720
  by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1721
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1722
lemma (in measure_space) integral_eq_positive_integral:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1723
  assumes f: "\<And>x. 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1724
  shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1725
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1726
  { fix x have "max 0 (extreal (- f x)) = 0" using f[of x] by (simp split: split_max) }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1727
  then have "0 = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1728
  also have "\<dots> = (\<integral>\<^isup>+ x. extreal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1729
  finally show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1730
    unfolding lebesgue_integral_def by simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1731
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1732
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1733
lemma (in measure_space) integral_vimage:
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1734
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1735
  assumes f: "f \<in> borel_measurable M'"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1736
  shows "integral\<^isup>L M' f = (\<integral>x. f (T x) \<partial>M)"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1737
proof -
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1738
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1739
  from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1740
  have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
41661
baf1964bc468 use pre-image measure, instead of image
hoelzl
parents: 41545
diff changeset
  1741
    and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1742
    using f by (auto simp: comp_def)
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1743
  then 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
  1744
    using f unfolding lebesgue_integral_def integrable_def
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1745
    by (auto simp: borel[THEN positive_integral_vimage[OF T]])
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1746
qed
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1747
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1748
lemma (in measure_space) integrable_vimage:
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1749
  assumes T: "sigma_algebra M'" "T \<in> measure_preserving M M'"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1750
  assumes f: "integrable M' f"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1751
  shows "integrable M (\<lambda>x. f (T x))"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1752
proof -
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1753
  interpret T: measure_space M' by (rule measure_space_vimage[OF T])
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1754
  from measurable_comp[OF measure_preservingD2[OF T(2)], of f borel]
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1755
  have borel: "(\<lambda>x. extreal (f x)) \<in> borel_measurable M'" "(\<lambda>x. extreal (- f x)) \<in> borel_measurable M'"
41831
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1756
    and "(\<lambda>x. f (T x)) \<in> borel_measurable M"
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1757
    using f by (auto simp: comp_def)
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1758
  then show ?thesis
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1759
    using f unfolding lebesgue_integral_def integrable_def
91a2b435dd7a use measure_preserving in ..._vimage lemmas
hoelzl
parents: 41705
diff changeset
  1760
    by (auto simp: borel[THEN positive_integral_vimage[OF T]])
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1761
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1762
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1763
lemma (in measure_space) integral_minus[intro, simp]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1764
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1765
  shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1766
  using assms by (auto simp: integrable_def lebesgue_integral_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1767
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1768
lemma (in measure_space) integral_of_positive_diff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1769
  assumes integrable: "integrable M u" "integrable M v"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1770
  and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
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
  1771
  shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1772
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1773
  let "?f x" = "max 0 (extreal (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1774
  let "?mf x" = "max 0 (extreal (- f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1775
  let "?u x" = "max 0 (extreal (u x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1776
  let "?v x" = "max 0 (extreal (v x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1777
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1778
  from borel_measurable_diff[of u v] integrable
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1779
  have f_borel: "?f \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1780
    mf_borel: "?mf \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1781
    v_borel: "?v \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1782
    u_borel: "?u \<in> borel_measurable M" and
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1783
    "f \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1784
    by (auto simp: f_def[symmetric] integrable_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1785
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1786
  have "(\<integral>\<^isup>+ x. extreal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1787
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1788
  moreover have "(\<integral>\<^isup>+ x. extreal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1789
    using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1790
  ultimately show f: "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1791
    using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1792
    by (auto simp: integrable_def f_def positive_integral_max_0)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1793
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1794
  have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1795
    unfolding f_def using pos by (simp split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1796
  then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1797
  then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1798
      real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1799
    using positive_integral_add[OF u_borel _ mf_borel]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1800
    using positive_integral_add[OF v_borel _ f_borel]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1801
    by auto
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1802
  then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1803
    unfolding positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1804
    unfolding pos[THEN integral_eq_positive_integral]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1805
    using integrable f by (auto elim!: integrableE)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1806
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1807
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1808
lemma (in measure_space) integral_linear:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1809
  assumes "integrable M f" "integrable M g" and "0 \<le> a"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1810
  shows "integrable M (\<lambda>t. a * f t + g t)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1811
  and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1812
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1813
  let "?f x" = "max 0 (extreal (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1814
  let "?g x" = "max 0 (extreal (g x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1815
  let "?mf x" = "max 0 (extreal (- f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1816
  let "?mg x" = "max 0 (extreal (- g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1817
  let "?p t" = "max 0 (a * f t) + max 0 (g t)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1818
  let "?n t" = "max 0 (- (a * f t)) + max 0 (- g t)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1819
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1820
  from assms have linear:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1821
    "(\<integral>\<^isup>+ x. extreal a * ?f x + ?g x \<partial>M) = extreal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1822
    "(\<integral>\<^isup>+ x. extreal a * ?mf x + ?mg x \<partial>M) = extreal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1823
    by (auto intro!: positive_integral_linear simp: integrable_def)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1824
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1825
  have *: "(\<integral>\<^isup>+x. extreal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- ?n x) \<partial>M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1826
    using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1827
  have **: "\<And>x. extreal a * ?f x + ?g x = max 0 (extreal (?p x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1828
           "\<And>x. extreal a * ?mf x + ?mg x = max 0 (extreal (?n x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1829
    using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1830
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
  1831
  have "integrable M ?p" "integrable M ?n"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1832
      "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1833
    using linear assms unfolding integrable_def ** *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1834
    by (auto simp: positive_integral_max_0)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1835
  note diff = integral_of_positive_diff[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1836
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
  1837
  show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1838
  from assms linear show ?EQ
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1839
    unfolding diff(2) ** positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1840
    unfolding lebesgue_integral_def *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1841
    by (auto elim!: integrableE simp: field_simps)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1842
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1843
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1844
lemma (in measure_space) integral_add[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1845
  assumes "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1846
  shows "integrable M (\<lambda>t. f t + g t)"
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
  1847
  and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1848
  using assms integral_linear[where a=1] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1849
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1850
lemma (in measure_space) integral_zero[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1851
  shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1852
  unfolding integrable_def lebesgue_integral_def
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1853
  by (auto simp add: borel_measurable_const)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1854
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1855
lemma (in measure_space) integral_cmult[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1856
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1857
  shows "integrable M (\<lambda>t. a * f t)" (is ?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
  1858
  and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1859
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
  1860
  have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1861
  proof (cases rule: le_cases)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1862
    assume "0 \<le> a" show ?thesis
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1863
      using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1864
      by (simp add: integral_zero)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1865
  next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1866
    assume "a \<le> 0" hence "0 \<le> - a" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1867
    have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1868
    show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1869
        integral_minus(1)[of "\<lambda>t. - a * f t"]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1870
      unfolding * integral_zero by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1871
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1872
  thus ?P ?I by auto
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1873
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1874
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1875
lemma (in measure_space) integral_multc:
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
  1876
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1877
  shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c"
41096
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1878
  unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
843c40bbc379 integral over setprod
hoelzl
parents: 41095
diff changeset
  1879
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1880
lemma (in measure_space) integral_mono_AE:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1881
  assumes fg: "integrable M f" "integrable M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1882
  and mono: "AE t. f t \<le> g t"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1883
  shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1884
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1885
  have "AE x. extreal (f x) \<le> extreal (g x)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1886
    using mono by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1887
  moreover have "AE x. extreal (- g x) \<le> extreal (- f x)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1888
    using mono by auto
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1889
  ultimately show ?thesis using fg
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1890
    by (auto intro!: add_mono positive_integral_mono_AE real_of_extreal_positive_mono
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1891
             simp: positive_integral_positive lebesgue_integral_def diff_minus)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1892
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  1893
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1894
lemma (in measure_space) integral_mono:
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1895
  assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1896
  shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  1897
  using assms by (auto intro: integral_mono_AE)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1898
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1899
lemma (in measure_space) integral_diff[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1900
  assumes f: "integrable M f" and g: "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1901
  shows "integrable M (\<lambda>t. f t - g t)"
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
  1902
  and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1903
  using integral_add[OF f integral_minus(1)[OF g]]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1904
  unfolding diff_minus integral_minus(2)[OF g]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1905
  by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1906
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1907
lemma (in measure_space) integral_indicator[simp, intro]:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1908
  assumes "A \<in> sets M" and "\<mu> A \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1909
  shows "integral\<^isup>L M (indicator A) = real (\<mu> A)" (is ?int)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1910
  and "integrable M (indicator A)" (is ?able)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1911
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1912
  from `A \<in> sets M` have *:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1913
    "\<And>x. extreal (indicator A x) = indicator A x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1914
    "(\<integral>\<^isup>+x. extreal (- indicator A x) \<partial>M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1915
    by (auto split: split_indicator simp: positive_integral_0_iff_AE one_extreal_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1916
  show ?int ?able
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1917
    using assms unfolding lebesgue_integral_def integrable_def
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1918
    by (auto simp: * positive_integral_indicator borel_measurable_indicator)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1919
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1920
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1921
lemma (in measure_space) integral_cmul_indicator:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1922
  assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> \<mu> A \<noteq> \<infinity>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1923
  shows "integrable M (\<lambda>x. c * indicator A x)" (is ?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
  1924
  and "(\<integral>x. c * indicator A x \<partial>M) = c * real (\<mu> A)" (is ?I)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1925
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1926
  show ?P
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1927
  proof (cases "c = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1928
    case False with assms show ?thesis by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1929
  qed simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1930
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1931
  show ?I
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1932
  proof (cases "c = 0")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1933
    case False with assms show ?thesis by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1934
  qed simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1935
qed
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1936
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1937
lemma (in measure_space) integral_setsum[simp, intro]:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1938
  assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f 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
  1939
  shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^isup>L M (f i))" (is "?int 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
  1940
    and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1941
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1942
  have "?int S \<and> ?I S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1943
  proof (cases "finite S")
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1944
    assume "finite S"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1945
    from this assms show ?thesis by (induct S) simp_all
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1946
  qed simp
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1947
  thus "?int S" and "?I S" by auto
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1948
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  1949
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  1950
lemma (in measure_space) integrable_abs:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1951
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1952
  shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  1953
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1954
  from assms have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>)\<partial>M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1955
    "\<And>x. extreal \<bar>f x\<bar> = max 0 (extreal (f x)) + max 0 (extreal (- f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1956
    by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1957
  with assms show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1958
    by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1959
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1960
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1961
lemma (in measure_space) integral_subalgebra:
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1962
  assumes borel: "f \<in> borel_measurable N"
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
  1963
  and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> measure N A = \<mu> A" and sa: "sigma_algebra 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
  1964
  shows "integrable N f \<longleftrightarrow> integrable M f" (is ?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
  1965
    and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1966
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
  1967
  interpret N: measure_space N using measure_space_subalgebra[OF sa N] .
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1968
  have "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1969
       "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1970
    using borel by (auto intro!: positive_integral_subalgebra N sa)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1971
  moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1972
    using assms unfolding measurable_def by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1973
  ultimately show ?P ?I
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1974
    by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
41545
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1975
qed
9c869baf1c66 tuned formalization of subalgebra
hoelzl
parents: 41544
diff changeset
  1976
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1977
lemma (in measure_space) integrable_bound:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1978
  assumes "integrable M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1979
  and f: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1980
    "\<And>x. x \<in> space M \<Longrightarrow> \<bar>g x\<bar> \<le> f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1981
  assumes borel: "g \<in> borel_measurable M"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  1982
  shows "integrable M g"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1983
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1984
  have "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal \<bar>g x\<bar> \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1985
    by (auto intro!: positive_integral_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1986
  also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1987
    using f by (auto intro!: positive_integral_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1988
  also have "\<dots> < \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1989
    using `integrable M f` unfolding integrable_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1990
  finally have pos: "(\<integral>\<^isup>+ x. extreal (g x) \<partial>M) < \<infinity>" .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1991
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1992
  have "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. extreal (\<bar>g x\<bar>) \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1993
    by (auto intro!: positive_integral_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1994
  also have "\<dots> \<le> (\<integral>\<^isup>+ x. extreal (f x) \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1995
    using f by (auto intro!: positive_integral_mono)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1996
  also have "\<dots> < \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1997
    using `integrable M f` unfolding integrable_def by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  1998
  finally have neg: "(\<integral>\<^isup>+ x. extreal (- g x) \<partial>M) < \<infinity>" .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  1999
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2000
  from neg pos borel show ?thesis
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2001
    unfolding integrable_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2002
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2003
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2004
lemma (in measure_space) integrable_abs_iff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2005
  "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2006
  by (auto intro!: integrable_bound[where g=f] integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2007
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2008
lemma (in measure_space) integrable_max:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2009
  assumes int: "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2010
  shows "integrable M (\<lambda> x. max (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2011
proof (rule integrable_bound)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2012
  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2013
    using int by (simp add: integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2014
  show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2015
    using int unfolding integrable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2016
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2017
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2018
  show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>max (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2019
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2020
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2021
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2022
lemma (in measure_space) integrable_min:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2023
  assumes int: "integrable M f" "integrable M g"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2024
  shows "integrable M (\<lambda> x. min (f x) (g x))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2025
proof (rule integrable_bound)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2026
  show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2027
    using int by (simp add: integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2028
  show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2029
    using int unfolding integrable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2030
next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2031
  fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2032
  show "0 \<le> \<bar>f x\<bar> + \<bar>g x\<bar>" "\<bar>min (f x) (g x)\<bar> \<le> \<bar>f x\<bar> + \<bar>g x\<bar>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2033
    by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2034
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2035
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2036
lemma (in measure_space) integral_triangle_inequality:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2037
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2038
  shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2039
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
  2040
  have "\<bar>integral\<^isup>L M f\<bar> = max (integral\<^isup>L M f) (- integral\<^isup>L M f)" 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
  2041
  also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2042
      using assms integral_minus(2)[of f, symmetric]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2043
      by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2044
  finally show ?thesis .
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2045
qed
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2046
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2047
lemma (in measure_space) integral_positive:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2048
  assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f 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
  2049
  shows "0 \<le> integral\<^isup>L M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2050
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
  2051
  have "0 = (\<integral>x. 0 \<partial>M)" by (auto simp: integral_zero)
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
  2052
  also have "\<dots> \<le> integral\<^isup>L M f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2053
    using assms by (rule integral_mono[OF integral_zero(1)])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2054
  finally show ?thesis .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2055
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2056
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2057
lemma (in measure_space) integral_monotone_convergence_pos:
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
  2058
  assumes i: "\<And>i. integrable M (f i)" and mono: "\<And>x. mono (\<lambda>n. f n x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2059
  and pos: "\<And>x i. 0 \<le> f i x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2060
  and lim: "\<And>x. (\<lambda>i. f i x) ----> u 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
  2061
  and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> 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
  2062
  shows "integrable M u"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2063
  and "integral\<^isup>L M u = x"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2064
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2065
  { fix x have "0 \<le> u x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2066
      using mono pos[of 0 x] incseq_le[OF _ lim, of x 0]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2067
      by (simp add: mono_def incseq_def) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2068
  note pos_u = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2069
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2070
  have SUP_F: "\<And>x. (SUP n. extreal (f n x)) = extreal (u x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2071
    unfolding SUP_eq_LIMSEQ[OF mono] by (rule lim)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2072
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2073
  have borel_f: "\<And>i. (\<lambda>x. extreal (f i x)) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2074
    using i unfolding integrable_def by auto
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2075
  hence "(\<lambda>x. SUP i. extreal (f i x)) \<in> borel_measurable M"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2076
    by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2077
  hence borel_u: "u \<in> borel_measurable M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2078
    by (auto simp: borel_measurable_extreal_iff SUP_F)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2079
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2080
  hence [simp]: "\<And>i. (\<integral>\<^isup>+x. extreal (- f i x) \<partial>M) = 0" "(\<integral>\<^isup>+x. extreal (- u x) \<partial>M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2081
    using i borel_u pos pos_u by (auto simp: positive_integral_0_iff_AE integrable_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2082
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2083
  have integral_eq: "\<And>n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M) = extreal (integral\<^isup>L M (f n))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2084
    using i positive_integral_positive by (auto simp: extreal_real lebesgue_integral_def integrable_def)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2085
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
  2086
  have pos_integral: "\<And>n. 0 \<le> integral\<^isup>L M (f n)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2087
    using pos i by (auto simp: integral_positive)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2088
  hence "0 \<le> x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2089
    using LIMSEQ_le_const[OF ilim, of 0] by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2090
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2091
  from mono pos i have pI: "(\<integral>\<^isup>+ x. extreal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. extreal (f n x) \<partial>M))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2092
    by (auto intro!: positive_integral_monotone_convergence_SUP
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2093
      simp: integrable_def incseq_mono incseq_Suc_iff le_fun_def SUP_F[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2094
  also have "\<dots> = extreal x" unfolding integral_eq
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2095
  proof (rule SUP_eq_LIMSEQ[THEN iffD2])
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
  2096
    show "mono (\<lambda>n. integral\<^isup>L M (f n))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2097
      using mono i by (auto simp: mono_def intro!: integral_mono)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2098
    show "(\<lambda>n. integral\<^isup>L M (f n)) ----> x" using ilim .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2099
  qed
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
  2100
  finally show  "integrable M u" "integral\<^isup>L M u = x" using borel_u `0 \<le> 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
  2101
    unfolding integrable_def lebesgue_integral_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2102
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2103
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2104
lemma (in measure_space) integral_monotone_convergence:
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
  2105
  assumes f: "\<And>i. integrable M (f i)" and "mono f"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2106
  and lim: "\<And>x. (\<lambda>i. f i x) ----> u 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
  2107
  and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> 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
  2108
  shows "integrable M u"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2109
  and "integral\<^isup>L M u = x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2110
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
  2111
  have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2112
      using f by (auto intro!: integral_diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2113
  have 2: "\<And>x. mono (\<lambda>n. f n x - f 0 x)" using `mono f`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2114
      unfolding mono_def le_fun_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2115
  have 3: "\<And>x n. 0 \<le> f n x - f 0 x" using `mono f`
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2116
      unfolding mono_def le_fun_def by (auto simp: field_simps)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2117
  have 4: "\<And>x. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2118
    using lim by (auto intro!: LIMSEQ_diff)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2119
  have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^isup>L M (f 0)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2120
    using f ilim by (auto intro!: LIMSEQ_diff simp: integral_diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2121
  note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5]
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
  2122
  have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2123
    using diff(1) f by (rule integral_add(1))
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
  2124
  with diff(2) f show "integrable M u" "integral\<^isup>L M u = x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2125
    by (auto simp: integral_diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2126
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2127
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2128
lemma (in measure_space) integral_0_iff:
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2129
  assumes "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2130
  shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> \<mu> {x\<in>space M. f x \<noteq> 0} = 0"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2131
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2132
  have *: "(\<integral>\<^isup>+x. extreal (- \<bar>f x\<bar>) \<partial>M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2133
    using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2134
  have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2135
  hence "(\<lambda>x. extreal (\<bar>f x\<bar>)) \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2136
    "(\<integral>\<^isup>+ x. extreal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2137
  from positive_integral_0_iff[OF this(1)] this(2)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2138
  show ?thesis unfolding lebesgue_integral_def *
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2139
    using positive_integral_positive[of "\<lambda>x. extreal \<bar>f x\<bar>"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2140
    by (auto simp add: real_of_extreal_eq_0)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2141
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2142
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2143
lemma (in measure_space) positive_integral_PInf:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2144
  assumes f: "f \<in> borel_measurable M"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2145
  and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2146
  shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2147
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2148
  have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2149
    using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2150
  also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2151
    by (auto intro!: positive_integral_mono simp: indicator_def max_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2152
  finally have "\<infinity> * \<mu> (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2153
    by (simp add: positive_integral_max_0)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2154
  moreover have "0 \<le> \<mu> (f -` {\<infinity>} \<inter> space M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2155
    using f by (simp add: measurable_sets)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2156
  ultimately show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2157
    using assms by (auto split: split_if_asm)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2158
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2159
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2160
lemma (in measure_space) positive_integral_PInf_AE:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2161
  assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x. f x \<noteq> \<infinity>"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
  2162
proof (rule AE_I)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2163
  show "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2164
    by (rule positive_integral_PInf[OF assms])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2165
  show "f -` {\<infinity>} \<inter> space M \<in> sets M"
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
  2166
    using assms by (auto intro: borel_measurable_vimage)
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
  2167
qed auto
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
  2168
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2169
lemma (in measure_space) simple_integral_PInf:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2170
  assumes "simple_function M f" "\<And>x. 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2171
  and "integral\<^isup>S M f \<noteq> \<infinity>"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2172
  shows "\<mu> (f -` {\<infinity>} \<inter> space M) = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2173
proof (rule positive_integral_PInf)
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2174
  show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2175
  show "integral\<^isup>P M f \<noteq> \<infinity>"
40859
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2176
    using assms by (simp add: positive_integral_eq_simple_integral)
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2177
qed
de0b30e6c2d2 Support product spaces on sigma finite measures.
hoelzl
parents: 40786
diff changeset
  2178
41026
bea75746dc9d folding on arbitrary Lebesgue integrable functions
hoelzl
parents: 41023
diff changeset
  2179
lemma (in measure_space) integral_real:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2180
  "AE x. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2181
  using assms unfolding lebesgue_integral_def
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2182
  by (subst (1 2) positive_integral_cong_AE) (auto simp add: extreal_real)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2183
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2184
lemma (in measure_space) integral_dominated_convergence:
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
  2185
  assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>x j. x\<in>space M \<Longrightarrow> \<bar>u j x\<bar> \<le> w x"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2186
  and w: "integrable M w"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2187
  and u': "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. u i x) ----> u' 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
  2188
  shows "integrable M u'"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2189
  and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2190
  and "(\<lambda>i. integral\<^isup>L M (u i)) ----> integral\<^isup>L M u'" (is ?lim)
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2191
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2192
  { fix x j assume x: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2193
    from u'[OF x] have "(\<lambda>i. \<bar>u i x\<bar>) ----> \<bar>u' x\<bar>" by (rule LIMSEQ_imp_rabs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2194
    from LIMSEQ_le_const2[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2195
    have "\<bar>u' x\<bar> \<le> w x" using bound[OF x] by auto }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2196
  note u'_bound = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2197
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2198
  from u[unfolded integrable_def]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2199
  have u'_borel: "u' \<in> borel_measurable M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2200
    using u' by (blast intro: borel_measurable_LIMSEQ[of u])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2201
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2202
  { fix x assume x: "x \<in> space M"
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2203
    then have "0 \<le> \<bar>u 0 x\<bar>" by auto
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2204
    also have "\<dots> \<le> w x" using bound[OF x] by auto
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2205
    finally have "0 \<le> w x" . }
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2206
  note w_pos = this
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2207
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2208
  show "integrable M u'"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2209
  proof (rule integrable_bound)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2210
    show "integrable M w" by fact
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2211
    show "u' \<in> borel_measurable M" by fact
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2212
  next
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2213
    fix x assume x: "x \<in> space M" then show "0 \<le> w x" by fact
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2214
    show "\<bar>u' x\<bar> \<le> w x" using u'_bound[OF x] .
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2215
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2216
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2217
  let "?diff n x" = "2 * w x - \<bar>u n x - u' x\<bar>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2218
  have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
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
  2219
    using w u `integrable M u'`
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2220
    by (auto intro!: integral_add integral_diff integral_cmult integrable_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2221
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2222
  { fix j x assume x: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2223
    have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2224
    also have "\<dots> \<le> w x + w x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2225
      by (rule add_mono[OF bound[OF x] u'_bound[OF x]])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2226
    finally have "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2227
  note diff_less_2w = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2228
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2229
  have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. extreal (?diff n x) \<partial>M) =
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2230
    (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2231
    using diff w diff_less_2w w_pos
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2232
    by (subst positive_integral_diff[symmetric])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2233
       (auto simp: integrable_def intro!: positive_integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2234
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
  2235
  have "integrable M (\<lambda>x. 2 * w x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2236
    using w by (auto intro: integral_cmult)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2237
  hence I2w_fin: "(\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2238
    borel_2w: "(\<lambda>x. extreal (2 * w x)) \<in> borel_measurable M"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2239
    unfolding integrable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2240
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2241
  have "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2242
  proof cases
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2243
    assume eq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2244
    { fix n
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2245
      have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2246
        using diff_less_2w[of _ n] unfolding positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2247
        by (intro positive_integral_mono) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2248
      then have "?f n = 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2249
        using positive_integral_positive[of ?f'] eq_0 by auto }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2250
    then show ?thesis by (simp add: Limsup_const)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2251
  next
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2252
    assume neq_0: "(\<integral>\<^isup>+ x. max 0 (extreal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2253
    have "0 = limsup (\<lambda>n. 0 :: extreal)" by (simp add: Limsup_const)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2254
    also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2255
      by (intro limsup_mono positive_integral_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2256
    finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)" .
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2257
    have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (extreal (?diff n x))) \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2258
    proof (rule positive_integral_cong)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2259
      fix x assume x: "x \<in> space M"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2260
      show "max 0 (extreal (2 * w x)) = liminf (\<lambda>n. max 0 (extreal (?diff n x)))"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2261
        unfolding extreal_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2262
      proof (rule lim_imp_Liminf[symmetric], unfold lim_extreal)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2263
        have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2264
          using u'[OF x] by (safe intro!: LIMSEQ_diff LIMSEQ_const LIMSEQ_imp_rabs)
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2265
        then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2266
          by (auto intro!: tendsto_real_max simp add: lim_extreal)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2267
      qed (rule trivial_limit_sequentially)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2268
    qed
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2269
    also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (extreal (?diff n x)) \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2270
      using u'_borel w u unfolding integrable_def
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2271
      by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2272
    also have "\<dots> = (\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M) -
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2273
        limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2274
      unfolding PI_diff positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2275
      using positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2276
      by (subst liminf_extreal_cminus) auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2277
    finally show ?thesis
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2278
      using neq_0 I2w_fin positive_integral_positive[of "\<lambda>x. extreal (2 * w x)"] pos
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2279
      unfolding positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2280
      by (cases rule: extreal2_cases[of "\<integral>\<^isup>+ x. extreal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M)"])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2281
         auto
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2282
  qed
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2283
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2284
  have "liminf ?f \<le> limsup ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2285
    by (intro extreal_Liminf_le_Limsup trivial_limit_sequentially)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2286
  moreover
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2287
  { have "0 = liminf (\<lambda>n. 0 :: extreal)" by (simp add: Liminf_const)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2288
    also have "\<dots> \<le> liminf ?f"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2289
      by (intro liminf_mono positive_integral_positive)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2290
    finally have "0 \<le> liminf ?f" . }
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2291
  ultimately have liminf_limsup_eq: "liminf ?f = extreal 0" "limsup ?f = extreal 0"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2292
    using `limsup ?f = 0` by auto
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2293
  have "\<And>n. (\<integral>\<^isup>+ x. extreal \<bar>u n x - u' x\<bar> \<partial>M) = extreal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2294
    using diff positive_integral_positive
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2295
    by (subst integral_eq_positive_integral) (auto simp: extreal_real integrable_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2296
  then show ?lim_diff
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2297
    using extreal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2298
    by (simp add: lim_extreal)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2299
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2300
  show ?lim
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2301
  proof (rule LIMSEQ_I)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2302
    fix r :: real assume "0 < r"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2303
    from LIMSEQ_D[OF `?lim_diff` this]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2304
    obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2305
      using diff by (auto simp: integral_positive)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2306
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
  2307
    show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2308
    proof (safe intro!: exI[of _ N])
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2309
      fix n assume "N \<le> n"
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
  2310
      have "\<bar>integral\<^isup>L M (u n) - integral\<^isup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
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
  2311
        using u `integrable M u'` by (auto simp: integral_diff)
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
  2312
      also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2313
        by (rule_tac integral_triangle_inequality) (auto intro!: integral_diff)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2314
      also note N[OF `N \<le> n`]
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
  2315
      finally show "norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2316
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2317
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2318
qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2319
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2320
lemma (in measure_space) integral_sums:
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
  2321
  assumes borel: "\<And>i. integrable M (f i)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2322
  and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2323
  and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2324
  shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?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
  2325
  and "(\<lambda>i. integral\<^isup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2326
proof -
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2327
  have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2328
    using summable unfolding summable_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2329
  from bchoice[OF this]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2330
  obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2331
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2332
  let "?w y" = "if y \<in> space M then w y else 0"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2333
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
  2334
  obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2335
    using sums unfolding summable_def ..
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2336
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
  2337
  have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2338
    using borel by (auto intro!: integral_setsum)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2339
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2340
  { fix j x assume [simp]: "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2341
    have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2342
    also have "\<dots> \<le> w x" using w[of x] series_pos_le[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2343
    finally have "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2344
  note 2 = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2345
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
  2346
  have 3: "integrable M ?w"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2347
  proof (rule integral_monotone_convergence(1))
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2348
    let "?F n y" = "(\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2349
    let "?w' n y" = "if y \<in> space M then ?F n y else 0"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2350
    have "\<And>n. integrable M (?F n)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2351
      using borel by (auto intro!: integral_setsum integrable_abs)
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2352
    thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2353
    show "mono ?w'"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2354
      by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2355
    { fix x show "(\<lambda>n. ?w' n x) ----> ?w x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2356
        using w by (cases "x \<in> space M") (simp_all add: LIMSEQ_const sums_def) }
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2357
    have *: "\<And>n. integral\<^isup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2358
      using borel by (simp add: integral_setsum integrable_abs cong: integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2359
    from abs_sum
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
  2360
    show "(\<lambda>i. integral\<^isup>L M (?w' i)) ----> x" unfolding * sums_def .
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2361
  qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2362
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2363
  from summable[THEN summable_rabs_cancel]
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2364
  have 4: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2365
    by (auto intro: summable_sumr_LIMSEQ_suminf)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2366
41705
1100512e16d8 add auto support for AE_mp
hoelzl
parents: 41689
diff changeset
  2367
  note int = integral_dominated_convergence(1,3)[OF 1 2 3 4]
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2368
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
  2369
  from int show "integrable M ?S" by simp
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2370
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2371
  show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2372
    using int(2) by simp
36624
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2373
qed
25153c08655e Cleanup information theory
hoelzl
parents: 35977
diff changeset
  2374
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2375
section "Lebesgue integration on countable spaces"
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2376
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2377
lemma (in measure_space) integral_on_countable:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2378
  assumes f: "f \<in> borel_measurable M"
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2379
  and bij: "bij_betw enum S (f ` space M)"
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2380
  and enum_zero: "enum ` (-S) \<subseteq> {0}"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2381
  and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2382
  and abs_summable: "summable (\<lambda>r. \<bar>enum r * real (\<mu> (f -` {enum r} \<inter> space M))\<bar>)"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2383
  shows "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2384
  and "(\<lambda>r. enum r * real (\<mu> (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2385
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2386
  let "?A r" = "f -` {enum r} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2387
  let "?F r x" = "enum r * indicator (?A r) 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
  2388
  have enum_eq: "\<And>r. enum r * real (\<mu> (?A r)) = integral\<^isup>L M (?F r)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2389
    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2390
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2391
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2392
    hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2393
    then obtain i where "i\<in>S" "enum i = f x" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2394
    have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2395
    proof cases
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2396
      fix j assume "j = i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2397
      thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2398
    next
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2399
      fix j assume "j \<noteq> i"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2400
      show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2401
        by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2402
    qed
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2403
    hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2404
    have "(\<lambda>i. ?F i x) sums f x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2405
         "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2406
      by (auto intro!: sums_single simp: F F_abs) }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2407
  note F_sums_f = this(1) and F_abs_sums_f = this(2)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2408
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
  2409
  have int_f: "integral\<^isup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2410
    using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2411
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2412
  { fix r
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2413
    have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2414
      by (auto simp: indicator_def intro!: integral_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2415
    also have "\<dots> = \<bar>enum r\<bar> * real (\<mu> (?A r))"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2416
      using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
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
  2417
    finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real (\<mu> (?A r))\<bar>"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2418
      using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_extreal_pos measurable_sets) }
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2419
  note int_abs_F = this
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2420
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
  2421
  have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2422
    using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2423
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2424
  have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2425
    using F_abs_sums_f unfolding sums_iff by auto
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2426
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2427
  from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2428
  show ?sums unfolding enum_eq int_f by simp
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2429
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2430
  from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2431
  show "integrable M f" unfolding int_f by simp
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2432
qed
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2433
35692
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  2434
section "Lebesgue integration on finite space"
f1315bbf1bc9 Moved theorems in Lebesgue to the right places
hoelzl
parents: 35582
diff changeset
  2435
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2436
lemma (in measure_space) integral_on_finite:
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2437
  assumes f: "f \<in> borel_measurable M" and finite: "finite (f`space M)"
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2438
  and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> \<mu> (f -` {x} \<inter> space M) \<noteq> \<infinity>"
41689
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2439
  shows "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2440
  and "(\<integral>x. f x \<partial>M) =
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2441
    (\<Sum> r \<in> f`space M. r * real (\<mu> (f -` {r} \<inter> space M)))" (is "?integral")
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2442
proof -
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2443
  let "?A r" = "f -` {r} \<inter> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2444
  let "?S x" = "\<Sum>r\<in>f`space M. r * indicator (?A r) x"
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2445
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2446
  { fix x assume "x \<in> space M"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2447
    have "f x = (\<Sum>r\<in>f`space M. if x \<in> ?A r then r else 0)"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2448
      using finite `x \<in> space M` by (simp add: setsum_cases)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2449
    also have "\<dots> = ?S x"
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2450
      by (auto intro!: setsum_cong)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2451
    finally have "f x = ?S x" . }
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2452
  note f_eq = this
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2453
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
  2454
  have f_eq_S: "integrable M f \<longleftrightarrow> integrable M ?S" "integral\<^isup>L M f = integral\<^isup>L M ?S"
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2455
    by (auto intro!: integrable_cong integral_cong simp only: f_eq)
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2456
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
  2457
  show "integrable M f" ?integral using fin f f_eq_S
38656
d5d342611edb Rewrite the Probability theory.
hoelzl
parents: 38642
diff changeset
  2458
    by (simp_all add: integral_cmul_indicator borel_measurable_vimage)
35582
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2459
qed
b16d99a72dc9 Add Lebesgue integral and probability space.
hoelzl
parents:
diff changeset
  2460
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
  2461
lemma (in finite_measure_space) simple_function_finite[simp, intro]: "simple_function M f"
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
  2462
  unfolding simple_function_def using finite_space by auto
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2463
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2464
lemma (in finite_measure_space) borel_measurable_finite[intro, simp]: "f \<in> borel_measurable M"
39092
98de40859858 move lemmas to correct theory files
hoelzl
parents: 38705
diff changeset
  2465
  by (auto intro: borel_measurable_simple_function)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2466
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2467
lemma (in finite_measure_space) positive_integral_finite_eq_setsum:
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2468
  assumes pos: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2469
  shows "integral\<^isup>P M f = (\<Sum>x \<in> space M. f x * \<mu> {x})"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2470
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
  2471
  have *: "integral\<^isup>P M f = (\<integral>\<^isup>+ x. (\<Sum>y\<in>space M. f y * indicator {y} x) \<partial>M)"
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2472
    by (auto intro!: positive_integral_cong simp add: indicator_def if_distrib setsum_cases[OF finite_space])
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2473
  show ?thesis unfolding * using borel_measurable_finite[of f] pos
40875
9a9d33f6fb46 generalized simple_functionD
hoelzl
parents: 40873
diff changeset
  2474
    by (simp add: positive_integral_setsum positive_integral_cmult_indicator)
38705
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2475
qed
aaee86c0e237 moved generic lemmas in Probability to HOL
hoelzl
parents: 38656
diff changeset
  2476
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2477
lemma (in finite_measure_space) integral_finite_singleton:
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
  2478
  shows "integrable M f"
3e39b0e730d6 the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents: 41661
diff changeset
  2479
  and "integral\<^isup>L M f = (\<Sum>x \<in> space M. f x * real (\<mu> {x}))" (is ?I)
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2480
proof -
41981
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2481
  have *:
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2482
    "(\<integral>\<^isup>+ x. max 0 (extreal (f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (f x)) * \<mu> {x})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2483
    "(\<integral>\<^isup>+ x. max 0 (extreal (- f x)) \<partial>M) = (\<Sum>x \<in> space M. max 0 (extreal (- f x)) * \<mu> {x})"
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2484
    by (simp_all add: positive_integral_finite_eq_setsum)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2485
  then show "integrable M f" using finite_space finite_measure
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2486
    by (simp add: setsum_Pinfty integrable_def positive_integral_max_0
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2487
             split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2488
  show ?I using finite_measure *
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2489
    apply (simp add: positive_integral_max_0 lebesgue_integral_def)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2490
    apply (subst (1 2) setsum_real_of_extreal[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2491
    apply (simp_all split: split_max add: setsum_subtractf[symmetric])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2492
    apply (intro setsum_cong[OF refl])
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2493
    apply (simp split: split_max)
cdf7693bbe08 reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents: 41831
diff changeset
  2494
    done
35977
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2495
qed
30d42bfd0174 Added finite measure space.
hoelzl
parents: 35833
diff changeset
  2496
35748
5f35613d9a65 Equality of integral and infinite sum.
hoelzl
parents: 35692
diff changeset
  2497
end