author | hoelzl |
Wed, 10 Oct 2012 12:12:31 +0200 | |
changeset 49795 | 9f2fb9b25a77 |
parent 49775 | 970964690b3d |
child 49796 | 182fa22e7ee8 |
permissions | -rw-r--r-- |
42067 | 1 |
(* 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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38656 | 5 |
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header {*Lebesgue Integration*} |
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||
38656 | 8 |
theory Lebesgue_Integration |
47694 | 9 |
imports Measure_Space Borel_Space |
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begin |
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||
47694 | 12 |
lemma ereal_minus_eq_PInfty_iff: |
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fixes x y :: ereal shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>" |
|
14 |
by (cases x y rule: ereal2_cases) simp_all |
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||
43920 | 16 |
lemma real_ereal_1[simp]: "real (1::ereal) = 1" |
17 |
unfolding one_ereal_def by simp |
|
42991
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integral strong monotone; finite subadditivity for measure
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parents:
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|
43920 | 19 |
lemma ereal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::ereal)" |
41981
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20 |
unfolding indicator_def by auto |
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|
cdf7693bbe08
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lemma tendsto_real_max: |
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fixes x y :: real |
cdf7693bbe08
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parents:
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|
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assumes "(X ---> x) net" |
cdf7693bbe08
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parents:
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25 |
assumes "(Y ---> y) net" |
cdf7693bbe08
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hoelzl
parents:
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|
26 |
shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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|
27 |
proof - |
cdf7693bbe08
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parents:
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|
28 |
have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2" |
cdf7693bbe08
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hoelzl
parents:
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|
29 |
by (auto split: split_max simp: field_simps) |
cdf7693bbe08
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parents:
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|
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show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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|
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unfolding * |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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|
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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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qed |
cdf7693bbe08
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parents:
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|
47694 | 35 |
lemma measurable_sets2[intro]: |
41981
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assumes "f \<in> measurable M M'" "g \<in> measurable M M''" |
cdf7693bbe08
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parents:
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and "A \<in> sets M'" "B \<in> sets M''" |
cdf7693bbe08
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hoelzl
parents:
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|
38 |
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:
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|
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proof - |
cdf7693bbe08
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hoelzl
parents:
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|
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have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)" |
cdf7693bbe08
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parents:
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|
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by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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|
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then show ?thesis using assms by (auto intro: measurable_sets) |
cdf7693bbe08
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parents:
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|
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qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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44 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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parents:
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|
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lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))" |
cdf7693bbe08
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|
46 |
proof |
cdf7693bbe08
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hoelzl
parents:
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|
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assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI) |
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parents:
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48 |
qed (auto simp: incseq_def) |
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parents:
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|
38656 | 50 |
section "Simple function" |
35582 | 51 |
|
38656 | 52 |
text {* |
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||
54 |
Our simple functions are not restricted to positive real numbers. Instead |
|
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they are just functions with a finite range and are measurable when singleton |
|
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sets are measurable. |
|
35582 | 57 |
|
38656 | 58 |
*} |
59 |
||
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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definition "simple_function M g \<longleftrightarrow> |
38656 | 61 |
finite (g ` space M) \<and> |
62 |
(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)" |
|
36624 | 63 |
|
47694 | 64 |
lemma simple_functionD: |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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assumes "simple_function M g" |
40875 | 66 |
shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M" |
40871 | 67 |
proof - |
68 |
show "finite (g ` space M)" |
|
69 |
using assms unfolding simple_function_def by auto |
|
40875 | 70 |
have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto |
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also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto |
|
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finally show "g -` X \<inter> space M \<in> sets M" using assms |
|
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by (auto intro!: finite_UN simp del: UN_simps simp: simple_function_def) |
|
40871 | 74 |
qed |
36624 | 75 |
|
47694 | 76 |
lemma simple_function_measurable2[intro]: |
41981
cdf7693bbe08
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hoelzl
parents:
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assumes "simple_function M f" "simple_function M g" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
78 |
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:
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diff
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|
79 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
80 |
have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)" |
cdf7693bbe08
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parents:
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|
81 |
by auto |
cdf7693bbe08
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parents:
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|
82 |
then show ?thesis using assms[THEN simple_functionD(2)] by auto |
cdf7693bbe08
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parents:
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|
83 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
84 |
|
47694 | 85 |
lemma simple_function_indicator_representation: |
43920 | 86 |
fixes f ::"'a \<Rightarrow> ereal" |
41689
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hoelzl
parents:
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|
87 |
assumes f: "simple_function M f" and x: "x \<in> space M" |
38656 | 88 |
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)" |
89 |
(is "?l = ?r") |
|
90 |
proof - |
|
38705 | 91 |
have "?r = (\<Sum>y \<in> f ` space M. |
38656 | 92 |
(if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))" |
93 |
by (auto intro!: setsum_cong2) |
|
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also have "... = f x * indicator (f -` {f x} \<inter> space M) x" |
|
95 |
using assms by (auto dest: simple_functionD simp: setsum_delta) |
|
96 |
also have "... = f x" using x by (auto simp: indicator_def) |
|
97 |
finally show ?thesis by auto |
|
98 |
qed |
|
36624 | 99 |
|
47694 | 100 |
lemma simple_function_notspace: |
43920 | 101 |
"simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h") |
35692 | 102 |
proof - |
38656 | 103 |
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto |
104 |
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset) |
|
105 |
have "?h -` {0} \<inter> space M = space M" by auto |
|
106 |
thus ?thesis unfolding simple_function_def by auto |
|
107 |
qed |
|
108 |
||
47694 | 109 |
lemma simple_function_cong: |
38656 | 110 |
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" |
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
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|
111 |
shows "simple_function M f \<longleftrightarrow> simple_function M g" |
38656 | 112 |
proof - |
113 |
have "f ` space M = g ` space M" |
|
114 |
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
115 |
using assms by (auto intro!: image_eqI) |
|
116 |
thus ?thesis unfolding simple_function_def using assms by simp |
|
117 |
qed |
|
118 |
||
47694 | 119 |
lemma simple_function_cong_algebra: |
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hoelzl
parents:
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120 |
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
|
121 |
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
|
122 |
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
|
123 |
|
47694 | 124 |
lemma borel_measurable_simple_function: |
41689
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hoelzl
parents:
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|
125 |
assumes "simple_function M f" |
38656 | 126 |
shows "f \<in> borel_measurable M" |
127 |
proof (rule borel_measurableI) |
|
128 |
fix S |
|
129 |
let ?I = "f ` (f -` S \<inter> space M)" |
|
130 |
have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto |
|
131 |
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
|
132 |
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
|
133 |
using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto |
38656 | 134 |
hence "?U \<in> sets M" |
135 |
apply (rule finite_UN) |
|
136 |
using assms unfolding simple_function_def by auto |
|
137 |
thus "f -` S \<inter> space M \<in> sets M" unfolding * . |
|
35692 | 138 |
qed |
139 |
||
47694 | 140 |
lemma simple_function_borel_measurable: |
41981
cdf7693bbe08
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hoelzl
parents:
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|
141 |
fixes f :: "'a \<Rightarrow> 'x::{t2_space}" |
38656 | 142 |
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
|
143 |
shows "simple_function M f" |
38656 | 144 |
using assms unfolding simple_function_def |
145 |
by (auto intro: borel_measurable_vimage) |
|
146 |
||
47694 | 147 |
lemma simple_function_eq_borel_measurable: |
43920 | 148 |
fixes f :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
149 |
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M" |
47694 | 150 |
using simple_function_borel_measurable[of f] borel_measurable_simple_function[of M f] |
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44666
diff
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|
151 |
by (fastforce simp: simple_function_def) |
41981
cdf7693bbe08
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hoelzl
parents:
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diff
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|
152 |
|
47694 | 153 |
lemma simple_function_const[intro, simp]: |
41689
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
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|
154 |
"simple_function M (\<lambda>x. c)" |
38656 | 155 |
by (auto intro: finite_subset simp: simple_function_def) |
47694 | 156 |
lemma simple_function_compose[intro, simp]: |
41689
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hoelzl
parents:
41661
diff
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|
157 |
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
|
158 |
shows "simple_function M (g \<circ> f)" |
38656 | 159 |
unfolding simple_function_def |
160 |
proof safe |
|
161 |
show "finite ((g \<circ> f) ` space M)" |
|
162 |
using assms unfolding simple_function_def by (auto simp: image_compose) |
|
163 |
next |
|
164 |
fix x assume "x \<in> space M" |
|
165 |
let ?G = "g -` {g (f x)} \<inter> (f`space M)" |
|
166 |
have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M = |
|
167 |
(\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto |
|
168 |
show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M" |
|
169 |
using assms unfolding simple_function_def * |
|
170 |
by (rule_tac finite_UN) (auto intro!: finite_UN) |
|
171 |
qed |
|
172 |
||
47694 | 173 |
lemma simple_function_indicator[intro, simp]: |
38656 | 174 |
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
|
175 |
shows "simple_function M (indicator A)" |
35692 | 176 |
proof - |
38656 | 177 |
have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _") |
178 |
by (auto simp: indicator_def) |
|
179 |
hence "finite ?S" by (rule finite_subset) simp |
|
180 |
moreover have "- A \<inter> space M = space M - A" by auto |
|
181 |
ultimately show ?thesis unfolding simple_function_def |
|
46905 | 182 |
using assms by (auto simp: indicator_def [abs_def]) |
35692 | 183 |
qed |
184 |
||
47694 | 185 |
lemma simple_function_Pair[intro, simp]: |
41689
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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 |
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
|
187 |
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
|
188 |
shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p") |
38656 | 189 |
unfolding simple_function_def |
190 |
proof safe |
|
191 |
show "finite (?p ` space M)" |
|
192 |
using assms unfolding simple_function_def |
|
193 |
by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto |
|
194 |
next |
|
195 |
fix x assume "x \<in> space M" |
|
196 |
have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = |
|
197 |
(f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" |
|
198 |
by auto |
|
199 |
with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M" |
|
200 |
using assms unfolding simple_function_def by auto |
|
201 |
qed |
|
35692 | 202 |
|
47694 | 203 |
lemma simple_function_compose1: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
204 |
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
|
205 |
shows "simple_function M (\<lambda>x. g (f x))" |
38656 | 206 |
using simple_function_compose[OF assms, of g] |
207 |
by (simp add: comp_def) |
|
35582 | 208 |
|
47694 | 209 |
lemma simple_function_compose2: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
210 |
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
|
211 |
shows "simple_function M (\<lambda>x. h (f x) (g x))" |
38656 | 212 |
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
|
213 |
have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))" |
38656 | 214 |
using assms by auto |
215 |
thus ?thesis by (simp_all add: comp_def) |
|
216 |
qed |
|
35582 | 217 |
|
47694 | 218 |
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"] |
38656 | 219 |
and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"] |
220 |
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"] |
|
221 |
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"] |
|
222 |
and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"] |
|
223 |
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"] |
|
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
224 |
and simple_function_max[intro, simp] = simple_function_compose2[where h=max] |
38656 | 225 |
|
47694 | 226 |
lemma simple_function_setsum[intro, simp]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
227 |
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
|
228 |
shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)" |
38656 | 229 |
proof cases |
230 |
assume "finite P" from this assms show ?thesis by induct auto |
|
231 |
qed auto |
|
35582 | 232 |
|
47694 | 233 |
lemma |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
234 |
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f" |
43920 | 235 |
shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
236 |
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
|
237 |
|
47694 | 238 |
lemma |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
239 |
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
|
240 |
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
|
241 |
by (auto intro!: simple_function_compose1[OF sf]) |
35582 | 242 |
|
47694 | 243 |
lemma borel_measurable_implies_simple_function_sequence: |
43920 | 244 |
fixes u :: "'a \<Rightarrow> ereal" |
38656 | 245 |
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
|
246 |
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
|
247 |
(\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)" |
35582 | 248 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
249 |
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
|
250 |
{ 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
|
251 |
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
|
252 |
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
|
253 |
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
|
254 |
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
|
255 |
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
|
256 |
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
|
257 |
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
|
258 |
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
|
259 |
qed auto } |
38656 | 260 |
note f_upper = this |
35582 | 261 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
262 |
have real_f: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
263 |
"\<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
|
264 |
unfolding f_def by auto |
35582 | 265 |
|
46731 | 266 |
let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
267 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
268 |
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
|
269 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
270 |
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
|
271 |
proof (intro simple_function_borel_measurable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
272 |
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
|
273 |
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
|
274 |
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
|
275 |
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
|
276 |
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
|
277 |
by (rule finite_subset) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
278 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
279 |
then show "simple_function M (?g i)" |
43920 | 280 |
by (auto intro: simple_function_ereal simple_function_div) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
281 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
282 |
show "incseq ?g" |
43920 | 283 |
proof (intro incseq_ereal incseq_SucI le_funI) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
284 |
fix x and i :: nat |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
285 |
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
|
286 |
proof ((split split_if)+, intro conjI impI) |
43920 | 287 |
assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
288 |
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
|
289 |
by (cases "u x") (auto intro!: le_natfloor) |
38656 | 290 |
next |
43920 | 291 |
assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
292 |
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
|
293 |
by (cases "u x") auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
294 |
next |
43920 | 295 |
assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
296 |
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
|
297 |
by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
298 |
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
|
299 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
300 |
assume "0 \<le> u x" then show ?thesis |
46671 | 301 |
by (intro le_mult_natfloor) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
302 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
303 |
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
|
304 |
by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg) |
38656 | 305 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
306 |
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
|
307 |
by (simp add: ac_simps) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
308 |
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
|
309 |
qed simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
310 |
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
|
311 |
by (auto simp: field_simps) |
35582 | 312 |
qed |
38656 | 313 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
314 |
fix x show "(SUP i. ?g i x) = max 0 (u x)" |
43920 | 315 |
proof (rule ereal_SUPI) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
316 |
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
|
317 |
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
|
318 |
mult_nonpos_nonneg mult_nonneg_nonneg) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
319 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
320 |
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
|
321 |
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
|
322 |
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
|
323 |
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
|
324 |
proof (cases y) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
325 |
case (real r) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
326 |
with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq) |
44666 | 327 |
from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le) |
43920 | 328 |
then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
329 |
then guess p .. note ux = this |
44666 | 330 |
obtain m :: nat where m: "p < real m" using reals_Archimedean2 .. |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
331 |
have "p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
332 |
proof (rule ccontr) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
333 |
assume "\<not> p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
334 |
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
|
335 |
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
|
336 |
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
|
337 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
338 |
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
|
339 |
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
|
340 |
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
|
341 |
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
|
342 |
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
|
343 |
ultimately show False by auto |
38656 | 344 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
345 |
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
|
346 |
qed (insert `0 \<le> y`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
347 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
348 |
qed (auto simp: divide_nonneg_pos) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
349 |
qed |
35582 | 350 |
|
47694 | 351 |
lemma borel_measurable_implies_simple_function_sequence': |
43920 | 352 |
fixes u :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
353 |
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
|
354 |
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
|
355 |
"\<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
|
356 |
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
|
357 |
|
47694 | 358 |
lemma simple_function_If_set: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
359 |
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
|
360 |
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
|
361 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
362 |
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
|
363 |
show ?thesis unfolding simple_function_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
364 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
365 |
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
|
366 |
from finite_subset[OF this] assms |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
367 |
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
|
368 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
369 |
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
|
370 |
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
|
371 |
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
|
372 |
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
|
373 |
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
|
374 |
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
|
375 |
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
|
376 |
show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto |
35582 | 377 |
qed |
378 |
qed |
|
379 |
||
47694 | 380 |
lemma simple_function_If: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
381 |
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
|
382 |
shows "simple_function M (\<lambda>x. if P x then f x else g x)" |
35582 | 383 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
384 |
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
|
385 |
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp |
38656 | 386 |
qed |
387 |
||
47694 | 388 |
lemma simple_function_subalgebra: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
389 |
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
|
390 |
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
|
391 |
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
|
392 |
using assms unfolding simple_function_def by auto |
39092 | 393 |
|
47694 | 394 |
lemma simple_function_comp: |
395 |
assumes T: "T \<in> measurable M M'" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
396 |
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
|
397 |
shows "simple_function M (\<lambda>x. f (T x))" |
41661 | 398 |
proof (intro simple_function_def[THEN iffD2] conjI ballI) |
399 |
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'" |
|
400 |
using T unfolding measurable_def by auto |
|
401 |
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
|
402 |
using f unfolding simple_function_def by (auto intro: finite_subset) |
41661 | 403 |
fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M" |
404 |
then have "i \<in> f ` space M'" |
|
405 |
using T unfolding measurable_def by auto |
|
406 |
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
|
407 |
using f unfolding simple_function_def by auto |
41661 | 408 |
then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M" |
409 |
using T unfolding measurable_def by auto |
|
410 |
also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M" |
|
411 |
using T unfolding measurable_def by auto |
|
412 |
finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" . |
|
40859 | 413 |
qed |
414 |
||
38656 | 415 |
section "Simple integral" |
416 |
||
47694 | 417 |
definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>S") where |
418 |
"integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
419 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
420 |
syntax |
47694 | 421 |
"_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<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
|
422 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
423 |
translations |
47694 | 424 |
"\<integral>\<^isup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)" |
35582 | 425 |
|
47694 | 426 |
lemma simple_integral_cong: |
38656 | 427 |
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
|
428 |
shows "integral\<^isup>S M f = integral\<^isup>S M g" |
38656 | 429 |
proof - |
430 |
have "f ` space M = g ` space M" |
|
431 |
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
432 |
using assms by (auto intro!: image_eqI) |
|
433 |
thus ?thesis unfolding simple_integral_def by simp |
|
434 |
qed |
|
435 |
||
47694 | 436 |
lemma simple_integral_const[simp]: |
437 |
"(\<integral>\<^isup>Sx. c \<partial>M) = c * (emeasure M) (space M)" |
|
38656 | 438 |
proof (cases "space M = {}") |
439 |
case True thus ?thesis unfolding simple_integral_def by simp |
|
440 |
next |
|
441 |
case False hence "(\<lambda>x. c) ` space M = {c}" by auto |
|
442 |
thus ?thesis unfolding simple_integral_def by simp |
|
35582 | 443 |
qed |
444 |
||
47694 | 445 |
lemma simple_function_partition: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
446 |
assumes f: "simple_function M f" and g: "simple_function M g" |
47694 | 447 |
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) * (emeasure M) A)" |
38656 | 448 |
(is "_ = setsum _ (?p ` space M)") |
449 |
proof- |
|
46731 | 450 |
let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)" |
38656 | 451 |
let ?SIGMA = "Sigma (f`space M) ?sub" |
35582 | 452 |
|
38656 | 453 |
have [intro]: |
454 |
"finite (f ` space M)" |
|
455 |
"finite (g ` space M)" |
|
456 |
using assms unfolding simple_function_def by simp_all |
|
457 |
||
458 |
{ fix A |
|
459 |
have "?p ` (A \<inter> space M) \<subseteq> |
|
460 |
(\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)" |
|
461 |
by auto |
|
462 |
hence "finite (?p ` (A \<inter> space M))" |
|
40786
0a54cfc9add3
gave more standard finite set rules simp and intro attribute
nipkow
parents:
39910
diff
changeset
|
463 |
by (rule finite_subset) auto } |
38656 | 464 |
note this[intro, simp] |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
465 |
note sets = simple_function_measurable2[OF f g] |
35582 | 466 |
|
38656 | 467 |
{ fix x assume "x \<in> space M" |
468 |
have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto |
|
47694 | 469 |
with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))" |
47761 | 470 |
by (subst setsum_emeasure) (auto simp: disjoint_family_on_def) } |
47694 | 471 |
hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
472 |
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
|
473 |
by (subst setsum_Sigma[symmetric]) |
43920 | 474 |
(auto intro!: setsum_cong setsum_ereal_right_distrib) |
47694 | 475 |
also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)" |
38656 | 476 |
proof - |
477 |
have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI) |
|
39910 | 478 |
have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M |
38656 | 479 |
= (\<lambda>x. (f x, ?p x)) ` space M" |
480 |
proof safe |
|
481 |
fix x assume "x \<in> space M" |
|
39910 | 482 |
thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M" |
38656 | 483 |
by (auto intro!: image_eqI[of _ _ "?p x"]) |
484 |
qed auto |
|
485 |
thus ?thesis |
|
39910 | 486 |
apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI) |
38656 | 487 |
apply (rule_tac x="xa" in image_eqI) |
488 |
by simp_all |
|
489 |
qed |
|
490 |
finally show ?thesis . |
|
35582 | 491 |
qed |
492 |
||
47694 | 493 |
lemma simple_integral_add[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
494 |
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
|
495 |
shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g" |
35582 | 496 |
proof - |
38656 | 497 |
{ fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M" |
498 |
assume "x \<in> space M" |
|
499 |
hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}" |
|
500 |
"(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S" |
|
501 |
by auto } |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
502 |
with assms show ?thesis |
38656 | 503 |
unfolding |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
504 |
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
|
505 |
simple_function_partition[OF f g] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
506 |
simple_function_partition[OF g f] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
507 |
by (subst (3) Int_commute) |
43920 | 508 |
(auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong) |
35582 | 509 |
qed |
510 |
||
47694 | 511 |
lemma simple_integral_setsum[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
512 |
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
|
513 |
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
|
514 |
shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))" |
38656 | 515 |
proof cases |
516 |
assume "finite P" |
|
517 |
from this assms show ?thesis |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
518 |
by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg) |
38656 | 519 |
qed auto |
520 |
||
47694 | 521 |
lemma simple_integral_mult[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
522 |
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
|
523 |
shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f" |
38656 | 524 |
proof - |
47694 | 525 |
note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)] |
38656 | 526 |
{ fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M" |
527 |
assume "x \<in> space M" |
|
528 |
hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}" |
|
529 |
by auto } |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
530 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
531 |
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
|
532 |
simple_function_partition[OF f(1) mult] |
43920 | 533 |
by (subst setsum_ereal_right_distrib) |
534 |
(auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc) |
|
40871 | 535 |
qed |
536 |
||
47694 | 537 |
lemma simple_integral_mono_AE: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
538 |
assumes f: "simple_function M f" and g: "simple_function M g" |
47694 | 539 |
and mono: "AE x in M. 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
|
540 |
shows "integral\<^isup>S M f \<le> integral\<^isup>S M g" |
40859 | 541 |
proof - |
46731 | 542 |
let ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)" |
40859 | 543 |
have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x" |
544 |
"\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto |
|
545 |
show ?thesis |
|
546 |
unfolding * |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
547 |
simple_function_partition[OF f g] |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
548 |
simple_function_partition[OF g f] |
40859 | 549 |
proof (safe intro!: setsum_mono) |
550 |
fix x assume "x \<in> space M" |
|
551 |
then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto |
|
47694 | 552 |
show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)" |
40859 | 553 |
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
|
554 |
case True then show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
555 |
using * assms(1,2)[THEN simple_functionD(2)] |
43920 | 556 |
by (auto intro!: ereal_mult_right_mono) |
40859 | 557 |
next |
558 |
case False |
|
47694 | 559 |
obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0" |
40859 | 560 |
using mono by (auto elim!: AE_E) |
561 |
have "?S x \<subseteq> N" using N `x \<in> space M` False by auto |
|
40871 | 562 |
moreover have "?S x \<in> sets M" using assms |
563 |
by (rule_tac Int) (auto intro!: simple_functionD) |
|
47694 | 564 |
ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N" |
565 |
using `N \<in> sets M` by (auto intro!: emeasure_mono) |
|
566 |
moreover have "0 \<le> (emeasure M) (?S x)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
567 |
using assms(1,2)[THEN simple_functionD(2)] by auto |
47694 | 568 |
ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
569 |
then show ?thesis by simp |
40859 | 570 |
qed |
571 |
qed |
|
572 |
qed |
|
573 |
||
47694 | 574 |
lemma simple_integral_mono: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
575 |
assumes "simple_function M f" and "simple_function M g" |
38656 | 576 |
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
|
577 |
shows "integral\<^isup>S M f \<le> integral\<^isup>S M g" |
41705 | 578 |
using assms by (intro simple_integral_mono_AE) auto |
35582 | 579 |
|
47694 | 580 |
lemma simple_integral_cong_AE: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
581 |
assumes "simple_function M f" and "simple_function M g" |
47694 | 582 |
and "AE x in M. 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
|
583 |
shows "integral\<^isup>S M f = integral\<^isup>S M g" |
40859 | 584 |
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE) |
585 |
||
47694 | 586 |
lemma simple_integral_cong': |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
587 |
assumes sf: "simple_function M f" "simple_function M g" |
47694 | 588 |
and mea: "(emeasure M) {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
|
589 |
shows "integral\<^isup>S M f = integral\<^isup>S M g" |
40859 | 590 |
proof (intro simple_integral_cong_AE sf AE_I) |
47694 | 591 |
show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact |
40859 | 592 |
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M" |
593 |
using sf[THEN borel_measurable_simple_function] by auto |
|
594 |
qed simp |
|
595 |
||
47694 | 596 |
lemma simple_integral_indicator: |
38656 | 597 |
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
|
598 |
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
|
599 |
shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = |
47694 | 600 |
(\<Sum>x \<in> f ` space M. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))" |
38656 | 601 |
proof cases |
602 |
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
|
603 |
moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f" |
38656 | 604 |
by (auto intro!: simple_integral_cong) |
605 |
moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto |
|
606 |
ultimately show ?thesis by (simp add: simple_integral_def) |
|
607 |
next |
|
608 |
assume "A \<noteq> space M" |
|
609 |
then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto |
|
610 |
have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _") |
|
35582 | 611 |
proof safe |
38656 | 612 |
fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto |
613 |
next |
|
614 |
fix y assume "y \<in> A" thus "f y \<in> ?I ` space M" |
|
615 |
using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y]) |
|
616 |
next |
|
617 |
show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x]) |
|
35582 | 618 |
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
|
619 |
have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = |
47694 | 620 |
(\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))" |
38656 | 621 |
unfolding simple_integral_def I |
622 |
proof (rule setsum_mono_zero_cong_left) |
|
623 |
show "finite (f ` space M \<union> {0})" |
|
624 |
using assms(2) unfolding simple_function_def by auto |
|
625 |
show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}" |
|
626 |
using sets_into_space[OF assms(1)] by auto |
|
40859 | 627 |
have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" |
628 |
by (auto simp: image_iff) |
|
38656 | 629 |
thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}). |
47694 | 630 |
i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto |
38656 | 631 |
next |
632 |
fix x assume "x \<in> f`A \<union> {0}" |
|
633 |
hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A" |
|
634 |
by (auto simp: indicator_def split: split_if_asm) |
|
47694 | 635 |
thus "x * (emeasure M) (?I -` {x} \<inter> space M) = |
636 |
x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all |
|
38656 | 637 |
qed |
638 |
show ?thesis unfolding * |
|
639 |
using assms(2) unfolding simple_function_def |
|
640 |
by (auto intro!: setsum_mono_zero_cong_right) |
|
641 |
qed |
|
35582 | 642 |
|
47694 | 643 |
lemma simple_integral_indicator_only[simp]: |
38656 | 644 |
assumes "A \<in> sets M" |
47694 | 645 |
shows "integral\<^isup>S M (indicator A) = emeasure M A" |
38656 | 646 |
proof cases |
647 |
assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto |
|
648 |
thus ?thesis unfolding simple_integral_def using `space M = {}` by auto |
|
649 |
next |
|
43920 | 650 |
assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto |
38656 | 651 |
thus ?thesis |
47694 | 652 |
using simple_integral_indicator[OF assms simple_function_const[of _ 1]] |
38656 | 653 |
using sets_into_space[OF assms] |
47694 | 654 |
by (auto intro!: arg_cong[where f="(emeasure M)"]) |
38656 | 655 |
qed |
35582 | 656 |
|
47694 | 657 |
lemma simple_integral_null_set: |
658 |
assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_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
|
659 |
shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0" |
38656 | 660 |
proof - |
47694 | 661 |
have "AE x in M. indicator N x = (0 :: ereal)" |
662 |
using `N \<in> null_sets M` 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
|
663 |
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
|
664 |
using assms apply (intro simple_integral_cong_AE) by auto |
40859 | 665 |
then show ?thesis by simp |
38656 | 666 |
qed |
35582 | 667 |
|
47694 | 668 |
lemma simple_integral_cong_AE_mult_indicator: |
669 |
assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<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
|
670 |
shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)" |
41705 | 671 |
using assms by (intro simple_integral_cong_AE) auto |
35582 | 672 |
|
47694 | 673 |
lemma simple_integral_distr: |
674 |
assumes T: "T \<in> measurable M M'" |
|
675 |
and f: "simple_function M' f" |
|
676 |
shows "integral\<^isup>S (distr M M' T) f = (\<integral>\<^isup>S x. f (T x) \<partial>M)" |
|
39092 | 677 |
unfolding simple_integral_def |
47694 | 678 |
proof (intro setsum_mono_zero_cong_right ballI) |
679 |
show "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space (distr M M' T)" |
|
680 |
using T unfolding measurable_def by auto |
|
681 |
show "finite (f ` space (distr M M' T))" |
|
682 |
using f unfolding simple_function_def by auto |
|
39092 | 683 |
next |
47694 | 684 |
fix i assume "i \<in> f ` space (distr M M' T) - (\<lambda>x. f (T x)) ` space M" |
685 |
then have "T -` (f -` {i} \<inter> space (distr M M' T)) \<inter> space M = {}" by (auto simp: image_iff) |
|
686 |
with f[THEN simple_functionD(2), of "{i}"] |
|
687 |
show "i * emeasure (distr M M' T) (f -` {i} \<inter> space (distr M M' T)) = 0" |
|
688 |
using T by (simp add: emeasure_distr) |
|
39092 | 689 |
next |
47694 | 690 |
fix i assume "i \<in> (\<lambda>x. f (T x)) ` space M" |
691 |
then have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M" |
|
692 |
using T unfolding measurable_def by auto |
|
693 |
with f[THEN simple_functionD(2), of "{i}"] T |
|
694 |
show "i * emeasure (distr M M' T) (f -` {i} \<inter> space (distr M M' T)) = |
|
695 |
i * (emeasure M) ((\<lambda>x. f (T x)) -` {i} \<inter> space M)" |
|
696 |
by (auto simp add: emeasure_distr) |
|
39092 | 697 |
qed |
698 |
||
47694 | 699 |
lemma simple_integral_cmult_indicator: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
700 |
assumes A: "A \<in> sets M" |
47694 | 701 |
shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * (emeasure M) A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
702 |
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
|
703 |
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
|
704 |
|
47694 | 705 |
lemma simple_integral_positive: |
706 |
assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
707 |
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
|
708 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
709 |
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
|
710 |
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
|
711 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
712 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
713 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
714 |
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
|
715 |
|
47694 | 716 |
definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>P") where |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
717 |
"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 | 718 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
719 |
syntax |
47694 | 720 |
"_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<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
|
721 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
722 |
translations |
47694 | 723 |
"\<integral>\<^isup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)" |
40872 | 724 |
|
47694 | 725 |
lemma positive_integral_positive: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
726 |
"0 \<le> integral\<^isup>P M f" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
727 |
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def) |
40873 | 728 |
|
47694 | 729 |
lemma positive_integral_not_MInfty[simp]: "integral\<^isup>P M f \<noteq> -\<infinity>" |
730 |
using positive_integral_positive[of M f] by auto |
|
731 |
||
732 |
lemma positive_integral_def_finite: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
733 |
"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
|
734 |
(is "_ = SUPR ?A ?f") |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
735 |
unfolding positive_integral_def |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
736 |
proof (safe intro!: antisym SUP_least) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
737 |
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
|
738 |
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
|
739 |
note gM = g(1)[THEN borel_measurable_simple_function] |
47694 | 740 |
have \<mu>G_pos: "0 \<le> (emeasure M) ?G" using gM by auto |
46731 | 741 |
let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
742 |
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
|
743 |
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
|
744 |
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
|
745 |
done |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
746 |
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
|
747 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
748 |
have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto |
47694 | 749 |
assume "(emeasure M) ?G = 0" |
750 |
with gM have "AE x in M. x \<notin> ?G" |
|
751 |
by (auto simp add: AE_iff_null intro!: null_setsI) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
752 |
with gM g show ?thesis |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
753 |
by (intro SUP_upper2[OF g0] simple_integral_mono_AE) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
754 |
(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
|
755 |
next |
47694 | 756 |
assume \<mu>G: "(emeasure M) ?G \<noteq> 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
757 |
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
|
758 |
proof (intro SUP_PInfty) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
759 |
fix n :: nat |
47694 | 760 |
let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)" |
43920 | 761 |
have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: ereal_divide_eq) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
762 |
then have "?g ?y \<in> ?A" by (rule g_in_A) |
47694 | 763 |
have "real n \<le> ?y * (emeasure M) ?G" |
764 |
using \<mu>G \<mu>G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
765 |
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
|
766 |
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
|
767 |
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
|
768 |
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
|
769 |
by (intro simple_integral_mono) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
770 |
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
|
771 |
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
|
772 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
773 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
774 |
qed |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
775 |
qed (auto intro: SUP_upper) |
40873 | 776 |
|
47694 | 777 |
lemma positive_integral_mono_AE: |
778 |
assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
779 |
unfolding positive_integral_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
780 |
proof (safe intro!: SUP_mono) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
781 |
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
|
782 |
from ae[THEN AE_E] guess N . note N = this |
47694 | 783 |
then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in) |
46731 | 784 |
let ?n = "\<lambda>x. n x * indicator (space M - N) x" |
47694 | 785 |
have "AE x in M. n x \<le> ?n x" "simple_function M ?n" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
786 |
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
|
787 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
788 |
{ 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
|
789 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
790 |
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
|
791 |
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
|
792 |
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
|
793 |
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
|
794 |
qed simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
795 |
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
|
796 |
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
|
797 |
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
|
798 |
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
|
799 |
by force |
38656 | 800 |
qed |
801 |
||
47694 | 802 |
lemma positive_integral_mono: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
803 |
"(\<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
|
804 |
by (auto intro: positive_integral_mono_AE) |
40859 | 805 |
|
47694 | 806 |
lemma positive_integral_cong_AE: |
807 |
"AE x in M. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v" |
|
40859 | 808 |
by (auto simp: eq_iff intro!: positive_integral_mono_AE) |
809 |
||
47694 | 810 |
lemma positive_integral_cong: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
811 |
"(\<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
|
812 |
by (auto intro: positive_integral_cong_AE) |
40859 | 813 |
|
47694 | 814 |
lemma positive_integral_eq_simple_integral: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
815 |
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
|
816 |
proof - |
46731 | 817 |
let ?f = "\<lambda>x. f x * indicator (space M) x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
818 |
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
|
819 |
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
|
820 |
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
|
821 |
have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f' |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
822 |
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
823 |
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
|
824 |
unfolding positive_integral_def |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
825 |
using f' by (auto intro!: SUP_upper) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
826 |
ultimately show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
827 |
by (simp cong: positive_integral_cong simple_integral_cong) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
828 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
829 |
|
47694 | 830 |
lemma positive_integral_eq_simple_integral_AE: |
831 |
assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
832 |
proof - |
47694 | 833 |
have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
834 |
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
|
835 |
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
|
836 |
add: positive_integral_eq_simple_integral) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
837 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
838 |
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
|
839 |
qed |
40873 | 840 |
|
47694 | 841 |
lemma positive_integral_SUP_approx: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
842 |
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
|
843 |
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
|
844 |
shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S") |
43920 | 845 |
proof (rule ereal_le_mult_one_interval) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
846 |
have "0 \<le> (SUP i. integral\<^isup>P M (f i))" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
847 |
using f(3) by (auto intro!: SUP_upper2 positive_integral_positive) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
848 |
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
|
849 |
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
|
850 |
using u(3) by auto |
43920 | 851 |
fix a :: ereal assume "0 < a" "a < 1" |
38656 | 852 |
hence "a \<noteq> 0" by auto |
46731 | 853 |
let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}" |
38656 | 854 |
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
|
855 |
using f `simple_function M u` by (auto simp: borel_measurable_simple_function) |
38656 | 856 |
|
46731 | 857 |
let ?uB = "\<lambda>i x. u x * indicator (?B i) x" |
38656 | 858 |
|
859 |
{ fix i have "?B i \<subseteq> ?B (Suc i)" |
|
860 |
proof safe |
|
861 |
fix i x assume "a * u x \<le> f i x" |
|
862 |
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
|
863 |
using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto |
38656 | 864 |
finally show "a * u x \<le> f (Suc i) x" . |
865 |
qed } |
|
866 |
note B_mono = this |
|
35582 | 867 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
868 |
note B_u = Int[OF u(1)[THEN simple_functionD(2)] B] |
38656 | 869 |
|
46731 | 870 |
let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n" |
47694 | 871 |
have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
872 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
873 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
874 |
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
|
875 |
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
|
876 |
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
|
877 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
878 |
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
|
879 |
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
|
880 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
881 |
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
|
882 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
883 |
assume "u x \<noteq> 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
884 |
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
|
885 |
have "a * u x < 1 * u x" |
43920 | 886 |
by (intro ereal_mult_strict_right_mono) (auto simp: image_iff) |
46884 | 887 |
also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def) |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
888 |
finally obtain i where "a * u x < f i x" unfolding SUP_def |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
889 |
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
|
890 |
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
|
891 |
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
|
892 |
qed |
40859 | 893 |
qed |
47694 | 894 |
then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
895 |
qed |
38656 | 896 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
897 |
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
|
898 |
unfolding simple_integral_indicator[OF B `simple_function M u`] |
43920 | 899 |
proof (subst SUPR_ereal_setsum, safe) |
38656 | 900 |
fix x n assume "x \<in> space M" |
47694 | 901 |
with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)" |
902 |
using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff) |
|
38656 | 903 |
next |
47694 | 904 |
show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
905 |
using measure_conv u_range B_u unfolding simple_integral_def |
43920 | 906 |
by (auto intro!: setsum_cong SUPR_ereal_cmult[symmetric]) |
38656 | 907 |
qed |
908 |
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
|
909 |
have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S" |
43920 | 910 |
apply (subst SUPR_ereal_cmult[symmetric]) |
38705 | 911 |
proof (safe intro!: SUP_mono bexI) |
38656 | 912 |
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
|
913 |
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
|
914 |
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
|
915 |
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
|
916 |
also have "\<dots> \<le> integral\<^isup>P M (f i)" |
38656 | 917 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
918 |
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
|
919 |
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
|
920 |
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
|
921 |
(auto intro!: positive_integral_mono split: split_indicator) |
38656 | 922 |
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
|
923 |
finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)" |
38656 | 924 |
by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
925 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
926 |
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
|
927 |
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
|
928 |
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
|
929 |
ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp |
35582 | 930 |
qed |
931 |
||
47694 | 932 |
lemma incseq_positive_integral: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
933 |
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
|
934 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
935 |
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
|
936 |
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
|
937 |
then show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
938 |
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
|
939 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
940 |
|
35582 | 941 |
text {* Beppo-Levi monotone convergence theorem *} |
47694 | 942 |
lemma positive_integral_monotone_convergence_SUP: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
943 |
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
|
944 |
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
|
945 |
proof (rule antisym) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
946 |
show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
947 |
by (auto intro!: SUP_least SUP_upper positive_integral_mono) |
38656 | 948 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
949 |
show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))" |
47694 | 950 |
unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"] |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
951 |
proof (safe intro!: SUP_least) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
952 |
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
|
953 |
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
|
954 |
moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
955 |
using f by (auto intro!: SUP_upper2) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
956 |
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
|
957 |
by (intro positive_integral_SUP_approx[OF f g _ g']) |
46884 | 958 |
(auto simp: le_fun_def max_def) |
35582 | 959 |
qed |
960 |
qed |
|
961 |
||
47694 | 962 |
lemma positive_integral_monotone_convergence_SUP_AE: |
963 |
assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
964 |
shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))" |
40859 | 965 |
proof - |
47694 | 966 |
from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
967 |
by (simp add: AE_all_countable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
968 |
from this[THEN AE_E] guess N . note N = this |
46731 | 969 |
let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0" |
47694 | 970 |
have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
971 |
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
|
972 |
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
|
973 |
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
|
974 |
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
|
975 |
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
|
976 |
{ 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
|
977 |
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
|
978 |
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
|
979 |
using N(1) by auto } |
40859 | 980 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
981 |
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
|
982 |
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
|
983 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
984 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
985 |
|
47694 | 986 |
lemma positive_integral_monotone_convergence_SUP_AE_incseq: |
987 |
assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
988 |
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
|
989 |
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
|
990 |
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
|
991 |
auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
992 |
|
47694 | 993 |
lemma positive_integral_monotone_convergence_simple: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
994 |
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
|
995 |
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
|
996 |
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
|
997 |
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
|
998 |
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
|
999 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1000 |
lemma positive_integral_max_0: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1001 |
"(\<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
|
1002 |
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
|
1003 |
|
47694 | 1004 |
lemma positive_integral_cong_pos: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1005 |
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
|
1006 |
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
|
1007 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1008 |
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
|
1009 |
proof (intro positive_integral_cong) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1010 |
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
|
1011 |
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
|
1012 |
by (auto split: split_max) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1013 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1014 |
then show ?thesis by (simp add: positive_integral_max_0) |
40859 | 1015 |
qed |
1016 |
||
47694 | 1017 |
lemma SUP_simple_integral_sequences: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1018 |
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
|
1019 |
and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)" |
47694 | 1020 |
and eq: "AE x in M. (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
|
1021 |
shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))" |
38656 | 1022 |
(is "SUPR _ ?F = SUPR _ ?G") |
1023 |
proof - |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1024 |
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
|
1025 |
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
|
1026 |
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
|
1027 |
unfolding eq[THEN positive_integral_cong_AE] .. |
38656 | 1028 |
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
|
1029 |
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
|
1030 |
finally show ?thesis by simp |
38656 | 1031 |
qed |
1032 |
||
47694 | 1033 |
lemma positive_integral_const[simp]: |
1034 |
"0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * (emeasure M) (space M)" |
|
38656 | 1035 |
by (subst positive_integral_eq_simple_integral) auto |
1036 |
||
47694 | 1037 |
lemma positive_integral_linear: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1038 |
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
|
1039 |
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
|
1040 |
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
|
1041 |
(is "integral\<^isup>P M ?L = _") |
35582 | 1042 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1043 |
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
|
1044 |
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
|
1045 |
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
|
1046 |
note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this |
46731 | 1047 |
let ?L' = "\<lambda>i x. a * u i x + v i x" |
38656 | 1048 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1049 |
have "?L \<in> borel_measurable M" using assms by auto |
38656 | 1050 |
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
|
1051 |
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
|
1052 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1053 |
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
|
1054 |
using u v `0 \<le> a` |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1055 |
by (auto simp: incseq_Suc_iff le_fun_def |
43920 | 1056 |
intro!: add_mono ereal_mult_left_mono simple_integral_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1057 |
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
|
1058 |
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
|
1059 |
{ 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
|
1060 |
by (auto split: split_if_asm) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1061 |
note not_MInf = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1062 |
|
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1063 |
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
|
1064 |
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
|
1065 |
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
|
1066 |
using u v `0 \<le> a` unfolding incseq_Suc_iff le_fun_def |
43920 | 1067 |
by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1068 |
{ fix x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1069 |
{ 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
|
1070 |
by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1071 |
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
|
1072 |
using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x] |
43920 | 1073 |
by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`]) |
1074 |
(auto intro!: SUPR_ereal_add |
|
1075 |
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) } |
|
47694 | 1076 |
then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1077 |
unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2) |
43920 | 1078 |
by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg) |
38656 | 1079 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1080 |
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
|
1081 |
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
|
1082 |
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
|
1083 |
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
|
1084 |
unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric] |
43920 | 1085 |
apply (subst SUPR_ereal_cmult[symmetric, OF pos(1) `0 \<le> a`]) |
1086 |
apply (subst SUPR_ereal_add[symmetric, OF inc not_MInf]) . |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1087 |
then show ?thesis by (simp add: positive_integral_max_0) |
38656 | 1088 |
qed |
1089 |
||
47694 | 1090 |
lemma positive_integral_cmult: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1091 |
assumes f: "f \<in> borel_measurable M" "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
|
1092 |
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
|
1093 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1094 |
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c` |
43920 | 1095 |
by (auto split: split_max simp: ereal_zero_le_0_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1096 |
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
|
1097 |
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
|
1098 |
then show ?thesis |
47694 | 1099 |
using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1100 |
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
|
1101 |
qed |
38656 | 1102 |
|
47694 | 1103 |
lemma positive_integral_multc: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1104 |
assumes "f \<in> borel_measurable M" "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
|
1105 |
shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c" |
41096 | 1106 |
unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp |
1107 |
||
47694 | 1108 |
lemma positive_integral_indicator[simp]: |
1109 |
"A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = (emeasure M) A" |
|
41544 | 1110 |
by (subst positive_integral_eq_simple_integral) |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1111 |
(auto simp: simple_integral_indicator) |
38656 | 1112 |
|
47694 | 1113 |
lemma positive_integral_cmult_indicator: |
1114 |
"0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A" |
|
41544 | 1115 |
by (subst positive_integral_eq_simple_integral) |
1116 |
(auto simp: simple_function_indicator simple_integral_indicator) |
|
38656 | 1117 |
|
47694 | 1118 |
lemma positive_integral_add: |
1119 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
1120 |
and g: "g \<in> borel_measurable M" "AE x in M. 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
|
1121 |
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
|
1122 |
proof - |
47694 | 1123 |
have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)" |
43920 | 1124 |
using assms by (auto split: split_max simp: ereal_add_nonneg_nonneg) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1125 |
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
|
1126 |
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
|
1127 |
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
|
1128 |
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
|
1129 |
also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)" |
47694 | 1130 |
using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1131 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1132 |
finally show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1133 |
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
|
1134 |
qed |
38656 | 1135 |
|
47694 | 1136 |
lemma positive_integral_setsum: |
1137 |
assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1138 |
shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))" |
38656 | 1139 |
proof cases |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1140 |
assume f: "finite P" |
47694 | 1141 |
from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1142 |
from f this assms(1) show ?thesis |
38656 | 1143 |
proof induct |
1144 |
case (insert i P) |
|
47694 | 1145 |
then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x" |
1146 |
"(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)" |
|
43920 | 1147 |
by (auto intro!: borel_measurable_ereal_setsum setsum_nonneg) |
38656 | 1148 |
from positive_integral_add[OF this] |
1149 |
show ?case using insert by auto |
|
1150 |
qed simp |
|
1151 |
qed simp |
|
1152 |
||
47694 | 1153 |
lemma positive_integral_Markov_inequality: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1154 |
assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" |
47694 | 1155 |
shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)" |
1156 |
(is "(emeasure M) ?A \<le> _ * ?PI") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1157 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1158 |
have "?A \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1159 |
using `A \<in> sets M` u by auto |
47694 | 1160 |
hence "(emeasure M) ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1161 |
using positive_integral_indicator by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1162 |
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
|
1163 |
by (auto intro!: positive_integral_mono_AE |
43920 | 1164 |
simp: indicator_def ereal_zero_le_0_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1165 |
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
|
1166 |
using assms |
43920 | 1167 |
by (auto intro!: positive_integral_cmult borel_measurable_indicator simp: ereal_zero_le_0_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1168 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1169 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1170 |
|
47694 | 1171 |
lemma positive_integral_noteq_infinite: |
1172 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1173 |
and "integral\<^isup>P M g \<noteq> \<infinity>" |
47694 | 1174 |
shows "AE x in M. g x \<noteq> \<infinity>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1175 |
proof (rule ccontr) |
47694 | 1176 |
assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)" |
1177 |
have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0" |
|
1178 |
using c g by (auto simp add: AE_iff_null) |
|
1179 |
moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets) |
|
1180 |
ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto |
|
1181 |
then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1182 |
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
|
1183 |
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
|
1184 |
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
|
1185 |
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
|
1186 |
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
|
1187 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1188 |
|
47694 | 1189 |
lemma positive_integral_diff: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1190 |
assumes f: "f \<in> borel_measurable M" |
47694 | 1191 |
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1192 |
and fin: "integral\<^isup>P M g \<noteq> \<infinity>" |
47694 | 1193 |
and mono: "AE x in M. 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
|
1194 |
shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g" |
38656 | 1195 |
proof - |
47694 | 1196 |
have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x" |
43920 | 1197 |
using assms by (auto intro: ereal_diff_positive) |
47694 | 1198 |
have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto |
43920 | 1199 |
{ fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b" |
1200 |
by (cases rule: ereal2_cases[of a b]) auto } |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1201 |
note * = this |
47694 | 1202 |
then have "AE x in M. f x = f x - g x + g x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1203 |
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
|
1204 |
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
|
1205 |
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
|
1206 |
by (rule positive_integral_cong_AE) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1207 |
show ?thesis unfolding ** |
47694 | 1208 |
using fin positive_integral_positive[of M g] |
43920 | 1209 |
by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto |
38656 | 1210 |
qed |
1211 |
||
47694 | 1212 |
lemma positive_integral_suminf: |
1213 |
assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1214 |
shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))" |
38656 | 1215 |
proof - |
47694 | 1216 |
have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1217 |
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
|
1218 |
have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))" |
43920 | 1219 |
using positive_integral_positive by (rule suminf_ereal_eq_SUPR) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1220 |
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
|
1221 |
unfolding positive_integral_setsum[OF f] .. |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1222 |
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
|
1223 |
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
|
1224 |
(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
|
1225 |
also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos |
43920 | 1226 |
by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUPR) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1227 |
finally show ?thesis by simp |
38656 | 1228 |
qed |
1229 |
||
1230 |
text {* Fatou's lemma: convergence theorem on limes inferior *} |
|
47694 | 1231 |
lemma positive_integral_lim_INF: |
43920 | 1232 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
47694 | 1233 |
assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1234 |
shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))" |
38656 | 1235 |
proof - |
47694 | 1236 |
have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1237 |
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
|
1238 |
(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
|
1239 |
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
|
1240 |
by (intro positive_integral_monotone_convergence_SUP_AE) |
44937
22c0857b8aab
removed further legacy rules from Complete_Lattices
hoelzl
parents:
44928
diff
changeset
|
1241 |
(elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1242 |
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
|
1243 |
unfolding liminf_SUPR_INFI |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
1244 |
by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower) |
38656 | 1245 |
finally show ?thesis . |
35582 | 1246 |
qed |
1247 |
||
47694 | 1248 |
lemma positive_integral_null_set: |
1249 |
assumes "N \<in> null_sets M" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0" |
|
38656 | 1250 |
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
|
1251 |
have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)" |
40859 | 1252 |
proof (intro positive_integral_cong_AE AE_I) |
1253 |
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N" |
|
1254 |
by (auto simp: indicator_def) |
|
47694 | 1255 |
show "(emeasure M) N = 0" "N \<in> sets M" |
40859 | 1256 |
using assms by auto |
35582 | 1257 |
qed |
40859 | 1258 |
then show ?thesis by simp |
38656 | 1259 |
qed |
35582 | 1260 |
|
47694 | 1261 |
lemma positive_integral_0_iff: |
1262 |
assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x" |
|
1263 |
shows "integral\<^isup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0" |
|
1264 |
(is "_ \<longleftrightarrow> (emeasure M) ?A = 0") |
|
35582 | 1265 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1266 |
have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u" |
38656 | 1267 |
by (auto intro!: positive_integral_cong simp: indicator_def) |
1268 |
show ?thesis |
|
1269 |
proof |
|
47694 | 1270 |
assume "(emeasure M) ?A = 0" |
1271 |
with positive_integral_null_set[of ?A M u] u |
|
1272 |
show "integral\<^isup>P M u = 0" by (simp add: u_eq null_sets_def) |
|
38656 | 1273 |
next |
43920 | 1274 |
{ fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r" |
1275 |
then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def) |
|
1276 |
then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) } |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1277 |
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
|
1278 |
assume *: "integral\<^isup>P M u = 0" |
46731 | 1279 |
let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}" |
47694 | 1280 |
have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))" |
38656 | 1281 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1282 |
{ fix n :: nat |
43920 | 1283 |
from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"] |
47694 | 1284 |
have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp |
1285 |
moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto |
|
1286 |
ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto } |
|
38656 | 1287 |
thus ?thesis by simp |
35582 | 1288 |
qed |
47694 | 1289 |
also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)" |
1290 |
proof (safe intro!: SUP_emeasure_incseq) |
|
38656 | 1291 |
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
|
1292 |
using u by (auto intro!: Int) |
38656 | 1293 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1294 |
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
|
1295 |
proof (safe intro!: incseq_SucI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1296 |
fix n :: nat and x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1297 |
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
|
1298 |
also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x" |
43920 | 1299 |
using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1300 |
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
|
1301 |
qed |
38656 | 1302 |
qed |
47694 | 1303 |
also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}" |
1304 |
proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1305 |
fix x assume "0 < u x" and [simp, intro]: "x \<in> space M" |
38656 | 1306 |
show "x \<in> (\<Union>n. ?M n \<inter> ?A)" |
1307 |
proof (cases "u x") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1308 |
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
|
1309 |
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
|
1310 |
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
|
1311 |
hence "1 \<le> real j * r" using real `0 < r` by auto |
43920 | 1312 |
thus ?thesis using `0 < r` real by (auto simp: one_ereal_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1313 |
qed (insert `0 < u x`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1314 |
qed auto |
47694 | 1315 |
finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1316 |
moreover |
47694 | 1317 |
from pos have "AE x in M. \<not> (u x < 0)" by auto |
1318 |
then have "(emeasure M) {x\<in>space M. u x < 0} = 0" |
|
1319 |
using AE_iff_null[of M] u by auto |
|
1320 |
moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}" |
|
1321 |
using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"]) |
|
1322 |
ultimately show "(emeasure M) ?A = 0" by simp |
|
35582 | 1323 |
qed |
1324 |
qed |
|
1325 |
||
47694 | 1326 |
lemma positive_integral_0_iff_AE: |
41705 | 1327 |
assumes u: "u \<in> borel_measurable M" |
47694 | 1328 |
shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)" |
41705 | 1329 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1330 |
have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M" |
41705 | 1331 |
using u by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1332 |
from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"] |
47694 | 1333 |
have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1334 |
unfolding positive_integral_max_0 |
47694 | 1335 |
using AE_iff_null[OF sets] u by auto |
1336 |
also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1337 |
finally show ?thesis . |
41705 | 1338 |
qed |
1339 |
||
47694 | 1340 |
lemma positive_integral_const_If: |
1341 |
"(\<integral>\<^isup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)" |
|
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1342 |
by (auto intro!: positive_integral_0_iff_AE[THEN iffD2]) |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1343 |
|
47694 | 1344 |
lemma positive_integral_subalgebra: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1345 |
assumes f: "f \<in> borel_measurable N" "AE x in N. 0 \<le> f x" |
47694 | 1346 |
and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1347 |
shows "integral\<^isup>P N f = integral\<^isup>P M f" |
39092 | 1348 |
proof - |
47694 | 1349 |
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess fs . note fs = this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1350 |
note sf = simple_function_subalgebra[OF fs(1) N(1,2)] |
47694 | 1351 |
from positive_integral_monotone_convergence_simple[OF fs(2,5,1), symmetric] |
1352 |
have "integral\<^isup>P N f = (SUP i. \<Sum>x\<in>fs i ` space M. x * emeasure N (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
|
1353 |
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
|
1354 |
unfolding simple_integral_def `space N = space M` by simp |
47694 | 1355 |
also have "\<dots> = (SUP i. \<Sum>x\<in>fs i ` space M. x * (emeasure M) (fs i -` {x} \<inter> space M))" |
1356 |
using 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
|
1357 |
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
|
1358 |
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
|
1359 |
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
|
1360 |
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
|
1361 |
finally show ?thesis . |
39092 | 1362 |
qed |
1363 |
||
35692 | 1364 |
section "Lebesgue Integral" |
1365 |
||
47694 | 1366 |
definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1367 |
"integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and> |
43920 | 1368 |
(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" |
35692 | 1369 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1370 |
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
|
1371 |
assumes "integrable M f" |
43920 | 1372 |
shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" |
38656 | 1373 |
using assms unfolding integrable_def by auto |
35692 | 1374 |
|
47694 | 1375 |
definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^isup>L") where |
43920 | 1376 |
"integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. ereal (- f x) \<partial>M))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1377 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1378 |
syntax |
47694 | 1379 |
"_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1380 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1381 |
translations |
47694 | 1382 |
"\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)" |
38656 | 1383 |
|
47694 | 1384 |
lemma integrableE: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1385 |
assumes "integrable M f" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1386 |
obtains r q where |
43920 | 1387 |
"(\<integral>\<^isup>+x. ereal (f x)\<partial>M) = ereal r" |
1388 |
"(\<integral>\<^isup>+x. ereal (-f x)\<partial>M) = ereal q" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1389 |
"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
|
1390 |
using assms unfolding integrable_def lebesgue_integral_def |
47694 | 1391 |
using positive_integral_positive[of M "\<lambda>x. ereal (f x)"] |
1392 |
using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"] |
|
43920 | 1393 |
by (cases rule: ereal2_cases[of "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. ereal (f x)\<partial>M)"]) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1394 |
|
47694 | 1395 |
lemma integral_cong: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1396 |
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
|
1397 |
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
|
1398 |
using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def) |
35582 | 1399 |
|
47694 | 1400 |
lemma integral_cong_AE: |
1401 |
assumes cong: "AE x in M. 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
|
1402 |
shows "integral\<^isup>L M f = integral\<^isup>L M g" |
40859 | 1403 |
proof - |
47694 | 1404 |
have *: "AE x in M. ereal (f x) = ereal (g x)" |
1405 |
"AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1406 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1407 |
unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def .. |
40859 | 1408 |
qed |
1409 |
||
47694 | 1410 |
lemma integrable_cong_AE: |
43339 | 1411 |
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
47694 | 1412 |
assumes "AE x in M. f x = g x" |
43339 | 1413 |
shows "integrable M f = integrable M g" |
1414 |
proof - |
|
43920 | 1415 |
have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (g x) \<partial>M)" |
1416 |
"(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (- g x) \<partial>M)" |
|
43339 | 1417 |
using assms by (auto intro!: positive_integral_cong_AE) |
1418 |
with assms show ?thesis |
|
1419 |
by (auto simp: integrable_def) |
|
1420 |
qed |
|
1421 |
||
47694 | 1422 |
lemma integrable_cong: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1423 |
"(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g" |
38656 | 1424 |
by (simp cong: positive_integral_cong measurable_cong add: integrable_def) |
1425 |
||
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1426 |
lemma integral_mono_AE: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1427 |
assumes fg: "integrable M f" "integrable M g" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1428 |
and mono: "AE t in M. f t \<le> g t" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1429 |
shows "integral\<^isup>L M f \<le> integral\<^isup>L M g" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1430 |
proof - |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1431 |
have "AE x in M. ereal (f x) \<le> ereal (g x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1432 |
using mono by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1433 |
moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1434 |
using mono by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1435 |
ultimately show ?thesis using fg |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1436 |
by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1437 |
simp: positive_integral_positive lebesgue_integral_def diff_minus) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1438 |
qed |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1439 |
|
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1440 |
lemma integral_mono: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1441 |
assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1442 |
shows "integral\<^isup>L M f \<le> integral\<^isup>L M g" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1443 |
using assms by (auto intro: integral_mono_AE) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1444 |
|
47694 | 1445 |
lemma positive_integral_eq_integral: |
1446 |
assumes f: "integrable M f" |
|
1447 |
assumes nonneg: "AE x in M. 0 \<le> f x" |
|
1448 |
shows "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = integral\<^isup>L M f" |
|
1449 |
proof - |
|
1450 |
have "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)" |
|
1451 |
using nonneg by (intro positive_integral_cong_AE) (auto split: split_max) |
|
1452 |
with f positive_integral_positive show ?thesis |
|
1453 |
by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>M") |
|
1454 |
(auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def) |
|
1455 |
qed |
|
1456 |
||
1457 |
lemma integral_eq_positive_integral: |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1458 |
assumes f: "\<And>x. 0 \<le> f x" |
43920 | 1459 |
shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)" |
35582 | 1460 |
proof - |
43920 | 1461 |
{ fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) } |
1462 |
then have "0 = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp |
|
1463 |
also have "\<dots> = (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 .. |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1464 |
finally show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1465 |
unfolding lebesgue_integral_def by simp |
35582 | 1466 |
qed |
1467 |
||
47694 | 1468 |
lemma integral_minus[intro, simp]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1469 |
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
|
1470 |
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
|
1471 |
using assms by (auto simp: integrable_def lebesgue_integral_def) |
38656 | 1472 |
|
47694 | 1473 |
lemma integral_minus_iff[simp]: |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1474 |
"integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1475 |
proof |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1476 |
assume "integrable M (\<lambda>x. - f x)" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1477 |
then have "integrable M (\<lambda>x. - (- f x))" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1478 |
by (rule integral_minus) |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1479 |
then show "integrable M f" by simp |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1480 |
qed (rule integral_minus) |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1481 |
|
47694 | 1482 |
lemma integral_of_positive_diff: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1483 |
assumes integrable: "integrable M u" "integrable M v" |
38656 | 1484 |
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
|
1485 |
shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v" |
35582 | 1486 |
proof - |
46731 | 1487 |
let ?f = "\<lambda>x. max 0 (ereal (f x))" |
1488 |
let ?mf = "\<lambda>x. max 0 (ereal (- f x))" |
|
1489 |
let ?u = "\<lambda>x. max 0 (ereal (u x))" |
|
1490 |
let ?v = "\<lambda>x. max 0 (ereal (v x))" |
|
38656 | 1491 |
|
47694 | 1492 |
from borel_measurable_diff[of u M v] integrable |
38656 | 1493 |
have f_borel: "?f \<in> borel_measurable M" and |
1494 |
mf_borel: "?mf \<in> borel_measurable M" and |
|
1495 |
v_borel: "?v \<in> borel_measurable M" and |
|
1496 |
u_borel: "?u \<in> borel_measurable M" and |
|
1497 |
"f \<in> borel_measurable M" |
|
1498 |
by (auto simp: f_def[symmetric] integrable_def) |
|
35582 | 1499 |
|
43920 | 1500 |
have "(\<integral>\<^isup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1501 |
using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0) |
43920 | 1502 |
moreover have "(\<integral>\<^isup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1503 |
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
|
1504 |
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
|
1505 |
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
|
1506 |
by (auto simp: integrable_def f_def positive_integral_max_0) |
35582 | 1507 |
|
38656 | 1508 |
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
|
1509 |
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
|
1510 |
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
|
1511 |
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
|
1512 |
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
|
1513 |
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
|
1514 |
using positive_integral_add[OF v_borel _ f_borel] |
38656 | 1515 |
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
|
1516 |
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
|
1517 |
unfolding positive_integral_max_0 |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1518 |
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
|
1519 |
using integrable f by (auto elim!: integrableE) |
35582 | 1520 |
qed |
1521 |
||
47694 | 1522 |
lemma integral_linear: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1523 |
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
|
1524 |
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
|
1525 |
and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ) |
38656 | 1526 |
proof - |
46731 | 1527 |
let ?f = "\<lambda>x. max 0 (ereal (f x))" |
1528 |
let ?g = "\<lambda>x. max 0 (ereal (g x))" |
|
1529 |
let ?mf = "\<lambda>x. max 0 (ereal (- f x))" |
|
1530 |
let ?mg = "\<lambda>x. max 0 (ereal (- g x))" |
|
1531 |
let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)" |
|
1532 |
let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)" |
|
38656 | 1533 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1534 |
from assms have linear: |
43920 | 1535 |
"(\<integral>\<^isup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g" |
1536 |
"(\<integral>\<^isup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1537 |
by (auto intro!: positive_integral_linear simp: integrable_def) |
35582 | 1538 |
|
43920 | 1539 |
have *: "(\<integral>\<^isup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. ereal (- ?n x) \<partial>M) = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1540 |
using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def) |
43920 | 1541 |
have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))" |
1542 |
"\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1543 |
using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff) |
35582 | 1544 |
|
41689
3e39b0e730d6
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 |
have "integrable M ?p" "integrable M ?n" |
38656 | 1546 |
"\<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
|
1547 |
using linear assms unfolding integrable_def ** * |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1548 |
by (auto simp: positive_integral_max_0) |
38656 | 1549 |
note diff = integral_of_positive_diff[OF this] |
1550 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1551 |
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
|
1552 |
from assms linear show ?EQ |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1553 |
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
|
1554 |
unfolding lebesgue_integral_def * |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1555 |
by (auto elim!: integrableE simp: field_simps) |
38656 | 1556 |
qed |
1557 |
||
47694 | 1558 |
lemma integral_add[simp, intro]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1559 |
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
|
1560 |
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
|
1561 |
and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g" |
38656 | 1562 |
using assms integral_linear[where a=1] by auto |
1563 |
||
47694 | 1564 |
lemma integral_zero[simp, intro]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1565 |
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
|
1566 |
unfolding integrable_def lebesgue_integral_def |
38656 | 1567 |
by (auto simp add: borel_measurable_const) |
35582 | 1568 |
|
47694 | 1569 |
lemma integral_cmult[simp, intro]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1570 |
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
|
1571 |
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
|
1572 |
and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I) |
38656 | 1573 |
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
|
1574 |
have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" |
38656 | 1575 |
proof (cases rule: le_cases) |
1576 |
assume "0 \<le> a" show ?thesis |
|
1577 |
using integral_linear[OF assms integral_zero(1) `0 \<le> a`] |
|
1578 |
by (simp add: integral_zero) |
|
1579 |
next |
|
1580 |
assume "a \<le> 0" hence "0 \<le> - a" by auto |
|
1581 |
have *: "\<And>t. - a * t + 0 = (-a) * t" by simp |
|
1582 |
show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`] |
|
47694 | 1583 |
integral_minus(1)[of M "\<lambda>t. - a * f t"] |
38656 | 1584 |
unfolding * integral_zero by simp |
1585 |
qed |
|
1586 |
thus ?P ?I by auto |
|
35582 | 1587 |
qed |
1588 |
||
47694 | 1589 |
lemma lebesgue_integral_cmult_nonneg: |
1590 |
assumes f: "f \<in> borel_measurable M" and "0 \<le> c" |
|
1591 |
shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f" |
|
1592 |
proof - |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1593 |
{ have "real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (f x)))) = |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1594 |
real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))" |
47694 | 1595 |
using f `0 \<le> c` by (subst positive_integral_cmult) auto |
1596 |
also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (c * f x))))" |
|
1597 |
using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff) |
|
1598 |
finally have "real (integral\<^isup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (f x)))" |
|
1599 |
by (simp add: positive_integral_max_0) } |
|
1600 |
moreover |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1601 |
{ have "real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (- f x)))) = |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1602 |
real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))" |
47694 | 1603 |
using f `0 \<le> c` by (subst positive_integral_cmult) auto |
1604 |
also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (- c * f x))))" |
|
1605 |
using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff) |
|
1606 |
finally have "real (integral\<^isup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (- f x)))" |
|
1607 |
by (simp add: positive_integral_max_0) } |
|
1608 |
ultimately show ?thesis |
|
1609 |
by (simp add: lebesgue_integral_def field_simps) |
|
1610 |
qed |
|
1611 |
||
1612 |
lemma lebesgue_integral_uminus: |
|
1613 |
"(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f" |
|
1614 |
unfolding lebesgue_integral_def by simp |
|
1615 |
||
1616 |
lemma lebesgue_integral_cmult: |
|
1617 |
assumes f: "f \<in> borel_measurable M" |
|
1618 |
shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f" |
|
1619 |
proof (cases rule: linorder_le_cases) |
|
1620 |
assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg) |
|
1621 |
next |
|
1622 |
assume "c \<le> 0" |
|
1623 |
with lebesgue_integral_cmult_nonneg[OF f, of "-c"] |
|
1624 |
show ?thesis |
|
1625 |
by (simp add: lebesgue_integral_def) |
|
1626 |
qed |
|
1627 |
||
1628 |
lemma 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
|
1629 |
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
|
1630 |
shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c" |
41096 | 1631 |
unfolding mult_commute[of _ c] integral_cmult[OF assms] .. |
1632 |
||
47694 | 1633 |
lemma integral_diff[simp, intro]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1634 |
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
|
1635 |
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
|
1636 |
and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g" |
38656 | 1637 |
using integral_add[OF f integral_minus(1)[OF g]] |
1638 |
unfolding diff_minus integral_minus(2)[OF g] |
|
1639 |
by auto |
|
1640 |
||
47694 | 1641 |
lemma integral_indicator[simp, intro]: |
1642 |
assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>" |
|
1643 |
shows "integral\<^isup>L M (indicator A) = real ((emeasure M) A)" (is ?int) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1644 |
and "integrable M (indicator A)" (is ?able) |
35582 | 1645 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1646 |
from `A \<in> sets M` have *: |
43920 | 1647 |
"\<And>x. ereal (indicator A x) = indicator A x" |
1648 |
"(\<integral>\<^isup>+x. ereal (- indicator A x) \<partial>M) = 0" |
|
1649 |
by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def) |
|
38656 | 1650 |
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
|
1651 |
using assms unfolding lebesgue_integral_def integrable_def |
38656 | 1652 |
by (auto simp: * positive_integral_indicator borel_measurable_indicator) |
35582 | 1653 |
qed |
1654 |
||
47694 | 1655 |
lemma integral_cmul_indicator: |
1656 |
assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1657 |
shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P) |
47694 | 1658 |
and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I) |
38656 | 1659 |
proof - |
1660 |
show ?P |
|
1661 |
proof (cases "c = 0") |
|
1662 |
case False with assms show ?thesis by simp |
|
1663 |
qed simp |
|
35582 | 1664 |
|
38656 | 1665 |
show ?I |
1666 |
proof (cases "c = 0") |
|
1667 |
case False with assms show ?thesis by simp |
|
1668 |
qed simp |
|
1669 |
qed |
|
35582 | 1670 |
|
47694 | 1671 |
lemma integral_setsum[simp, intro]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1672 |
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
|
1673 |
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
|
1674 |
and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S") |
35582 | 1675 |
proof - |
38656 | 1676 |
have "?int S \<and> ?I S" |
1677 |
proof (cases "finite S") |
|
1678 |
assume "finite S" |
|
1679 |
from this assms show ?thesis by (induct S) simp_all |
|
1680 |
qed simp |
|
35582 | 1681 |
thus "?int S" and "?I S" by auto |
1682 |
qed |
|
1683 |
||
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1684 |
lemma integrable_bound: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1685 |
assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1686 |
assumes borel: "g \<in> borel_measurable M" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1687 |
shows "integrable M g" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1688 |
proof - |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1689 |
have "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal \<bar>g x\<bar> \<partial>M)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1690 |
by (auto intro!: positive_integral_mono) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1691 |
also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1692 |
using f by (auto intro!: positive_integral_mono_AE) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1693 |
also have "\<dots> < \<infinity>" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1694 |
using `integrable M f` unfolding integrable_def by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1695 |
finally have pos: "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) < \<infinity>" . |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1696 |
|
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1697 |
have "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1698 |
by (auto intro!: positive_integral_mono) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1699 |
also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1700 |
using f by (auto intro!: positive_integral_mono_AE) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1701 |
also have "\<dots> < \<infinity>" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1702 |
using `integrable M f` unfolding integrable_def by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1703 |
finally have neg: "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) < \<infinity>" . |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1704 |
|
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1705 |
from neg pos borel show ?thesis |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1706 |
unfolding integrable_def by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1707 |
qed |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1708 |
|
47694 | 1709 |
lemma integrable_abs: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1710 |
assumes f: "integrable M f" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1711 |
shows "integrable M (\<lambda> x. \<bar>f x\<bar>)" |
36624 | 1712 |
proof - |
43920 | 1713 |
from assms have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0" |
1714 |
"\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1715 |
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
|
1716 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1717 |
by (simp add: positive_integral_add positive_integral_max_0 integrable_def) |
38656 | 1718 |
qed |
1719 |
||
47694 | 1720 |
lemma integral_subalgebra: |
41545 | 1721 |
assumes borel: "f \<in> borel_measurable N" |
47694 | 1722 |
and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1723 |
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
|
1724 |
and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I) |
41545 | 1725 |
proof - |
43920 | 1726 |
have "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M)" |
1727 |
"(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)" |
|
47694 | 1728 |
using borel by (auto intro!: positive_integral_subalgebra N) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1729 |
moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N" |
41545 | 1730 |
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
|
1731 |
ultimately show ?P ?I |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1732 |
by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0) |
41545 | 1733 |
qed |
1734 |
||
47694 | 1735 |
lemma lebesgue_integral_nonneg: |
1736 |
assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^isup>L M f" |
|
1737 |
proof - |
|
1738 |
have "(\<integral>\<^isup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+x. 0 \<partial>M)" |
|
1739 |
using ae by (intro positive_integral_cong_AE) (auto simp: max_def) |
|
1740 |
then show ?thesis |
|
1741 |
by (auto simp: lebesgue_integral_def positive_integral_max_0 |
|
1742 |
intro!: real_of_ereal_pos positive_integral_positive) |
|
1743 |
qed |
|
1744 |
||
1745 |
lemma integrable_abs_iff: |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1746 |
"f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f" |
38656 | 1747 |
by (auto intro!: integrable_bound[where g=f] integrable_abs) |
1748 |
||
47694 | 1749 |
lemma integrable_max: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1750 |
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
|
1751 |
shows "integrable M (\<lambda> x. max (f x) (g x))" |
38656 | 1752 |
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
|
1753 |
show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)" |
38656 | 1754 |
using int by (simp add: integrable_abs) |
1755 |
show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M" |
|
1756 |
using int unfolding integrable_def by auto |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1757 |
qed auto |
38656 | 1758 |
|
47694 | 1759 |
lemma integrable_min: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1760 |
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
|
1761 |
shows "integrable M (\<lambda> x. min (f x) (g x))" |
38656 | 1762 |
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
|
1763 |
show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)" |
38656 | 1764 |
using int by (simp add: integrable_abs) |
1765 |
show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M" |
|
1766 |
using int unfolding integrable_def by auto |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1767 |
qed auto |
38656 | 1768 |
|
47694 | 1769 |
lemma integral_triangle_inequality: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1770 |
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
|
1771 |
shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)" |
38656 | 1772 |
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
|
1773 |
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
|
1774 |
also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)" |
47694 | 1775 |
using assms integral_minus(2)[of M f, symmetric] |
38656 | 1776 |
by (auto intro!: integral_mono integrable_abs simp del: integral_minus) |
1777 |
finally show ?thesis . |
|
36624 | 1778 |
qed |
1779 |
||
47694 | 1780 |
lemma integral_positive: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1781 |
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
|
1782 |
shows "0 \<le> integral\<^isup>L M f" |
38656 | 1783 |
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
|
1784 |
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
|
1785 |
also have "\<dots> \<le> integral\<^isup>L M f" |
38656 | 1786 |
using assms by (rule integral_mono[OF integral_zero(1)]) |
1787 |
finally show ?thesis . |
|
1788 |
qed |
|
1789 |
||
47694 | 1790 |
lemma integral_monotone_convergence_pos: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1791 |
assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1792 |
and pos: "\<And>i. AE x in M. 0 \<le> f i x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1793 |
and lim: "AE x in M. (\<lambda>i. f i x) ----> u x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1794 |
and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1795 |
and u: "u \<in> borel_measurable M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1796 |
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
|
1797 |
and "integral\<^isup>L M u = x" |
35582 | 1798 |
proof - |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1799 |
have "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. ereal (f n x) \<partial>M))" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1800 |
proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric]) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1801 |
fix i |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1802 |
from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1803 |
by eventually_elim (auto simp: mono_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1804 |
show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1805 |
using i by (auto simp: integrable_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1806 |
next |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1807 |
show "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = \<integral>\<^isup>+ x. (SUP i. ereal (f i x)) \<partial>M" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1808 |
apply (rule positive_integral_cong_AE) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1809 |
using lim mono |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1810 |
by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2]) |
38656 | 1811 |
qed |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1812 |
also have "\<dots> = ereal x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1813 |
using mono i unfolding positive_integral_eq_integral[OF i pos] |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1814 |
by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1815 |
finally have "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = ereal x" . |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1816 |
moreover have "(\<integral>\<^isup>+ x. ereal (- u x) \<partial>M) = 0" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1817 |
proof (subst positive_integral_0_iff_AE) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1818 |
show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1819 |
using u by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1820 |
from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1821 |
proof eventually_elim |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1822 |
fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1823 |
then show "ereal (- u x) \<le> 0" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1824 |
using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1825 |
qed |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1826 |
qed |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1827 |
ultimately show "integrable M u" "integral\<^isup>L M u = x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1828 |
by (auto simp: integrable_def lebesgue_integral_def u) |
38656 | 1829 |
qed |
1830 |
||
47694 | 1831 |
lemma integral_monotone_convergence: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1832 |
assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1833 |
and lim: "AE x in M. (\<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
|
1834 |
and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1835 |
and u: "u \<in> borel_measurable M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1836 |
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
|
1837 |
and "integral\<^isup>L M u = x" |
38656 | 1838 |
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
|
1839 |
have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1840 |
using f by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1841 |
have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1842 |
using mono by (auto simp: mono_def le_fun_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1843 |
have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1844 |
using mono by (auto simp: field_simps mono_def le_fun_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1845 |
have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x" |
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
43941
diff
changeset
|
1846 |
using lim by (auto intro!: tendsto_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
|
1847 |
have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^isup>L M (f 0)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1848 |
using f ilim by (auto intro!: tendsto_diff) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1849 |
have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1850 |
using f[of 0] u by auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1851 |
note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6] |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1852 |
have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)" |
38656 | 1853 |
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
|
1854 |
with diff(2) f show "integrable M u" "integral\<^isup>L M u = x" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1855 |
by auto |
38656 | 1856 |
qed |
1857 |
||
47694 | 1858 |
lemma integral_0_iff: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1859 |
assumes "integrable M f" |
47694 | 1860 |
shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0" |
38656 | 1861 |
proof - |
43920 | 1862 |
have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1863 |
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
|
1864 |
have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs) |
43920 | 1865 |
hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M" |
1866 |
"(\<integral>\<^isup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto |
|
38656 | 1867 |
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
|
1868 |
show ?thesis unfolding lebesgue_integral_def * |
47694 | 1869 |
using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"] |
43920 | 1870 |
by (auto simp add: real_of_ereal_eq_0) |
35582 | 1871 |
qed |
1872 |
||
47694 | 1873 |
lemma positive_integral_PInf: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1874 |
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
|
1875 |
and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>" |
47694 | 1876 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
40859 | 1877 |
proof - |
47694 | 1878 |
have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1879 |
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
|
1880 |
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
|
1881 |
by (auto intro!: positive_integral_mono simp: indicator_def max_def) |
47694 | 1882 |
finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1883 |
by (simp add: positive_integral_max_0) |
47694 | 1884 |
moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)" |
1885 |
by (rule emeasure_nonneg) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1886 |
ultimately show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1887 |
using assms by (auto split: split_if_asm) |
40859 | 1888 |
qed |
1889 |
||
47694 | 1890 |
lemma positive_integral_PInf_AE: |
1891 |
assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>" |
|
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1892 |
proof (rule AE_I) |
47694 | 1893 |
show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1894 |
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
|
1895 |
show "f -` {\<infinity>} \<inter> space M \<in> sets M" |
41026
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1896 |
using assms by (auto intro: borel_measurable_vimage) |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1897 |
qed auto |
bea75746dc9d
folding on arbitrary Lebesgue integrable functions
hoelzl
parents:
41023
diff
changeset
|
1898 |
|
47694 | 1899 |
lemma simple_integral_PInf: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1900 |
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
|
1901 |
and "integral\<^isup>S M f \<noteq> \<infinity>" |
47694 | 1902 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1903 |
proof (rule positive_integral_PInf) |
40859 | 1904 |
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
|
1905 |
show "integral\<^isup>P M f \<noteq> \<infinity>" |
40859 | 1906 |
using assms by (simp add: positive_integral_eq_simple_integral) |
1907 |
qed |
|
1908 |
||
47694 | 1909 |
lemma integral_real: |
1910 |
"AE x in M. \<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))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1911 |
using assms unfolding lebesgue_integral_def |
43920 | 1912 |
by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1913 |
|
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1914 |
lemma (in finite_measure) lebesgue_integral_const[simp]: |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1915 |
shows "integrable M (\<lambda>x. a)" |
47694 | 1916 |
and "(\<integral>x. a \<partial>M) = a * (measure M) (space M)" |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1917 |
proof - |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1918 |
{ fix a :: real assume "0 \<le> a" |
47694 | 1919 |
then have "(\<integral>\<^isup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)" |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1920 |
by (subst positive_integral_const) auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1921 |
moreover |
43920 | 1922 |
from `0 \<le> a` have "(\<integral>\<^isup>+ x. ereal (-a) \<partial>M) = 0" |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1923 |
by (subst positive_integral_0_iff_AE) auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1924 |
ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) } |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1925 |
note * = this |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1926 |
show "integrable M (\<lambda>x. a)" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1927 |
proof cases |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1928 |
assume "0 \<le> a" with * show ?thesis . |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1929 |
next |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1930 |
assume "\<not> 0 \<le> a" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1931 |
then have "0 \<le> -a" by auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1932 |
from *[OF this] show ?thesis by simp |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1933 |
qed |
47694 | 1934 |
show "(\<integral>x. a \<partial>M) = a * measure M (space M)" |
1935 |
by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure) |
|
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1936 |
qed |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1937 |
|
43339 | 1938 |
lemma indicator_less[simp]: |
43920 | 1939 |
"indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)" |
43339 | 1940 |
by (simp add: indicator_def not_le) |
1941 |
||
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1942 |
lemma (in finite_measure) integral_less_AE: |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1943 |
assumes int: "integrable M X" "integrable M Y" |
47694 | 1944 |
assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x" |
1945 |
assumes gt: "AE x in M. X x \<le> Y x" |
|
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1946 |
shows "integral\<^isup>L M X < integral\<^isup>L M Y" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1947 |
proof - |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1948 |
have "integral\<^isup>L M X \<le> integral\<^isup>L M Y" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1949 |
using gt int by (intro integral_mono_AE) auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1950 |
moreover |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1951 |
have "integral\<^isup>L M X \<noteq> integral\<^isup>L M Y" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1952 |
proof |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1953 |
assume eq: "integral\<^isup>L M X = integral\<^isup>L M Y" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1954 |
have "integral\<^isup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^isup>L M (\<lambda>x. Y x - X x)" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1955 |
using gt by (intro integral_cong_AE) auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1956 |
also have "\<dots> = 0" |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1957 |
using eq int by simp |
47694 | 1958 |
finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0" |
43339 | 1959 |
using int by (simp add: integral_0_iff) |
1960 |
moreover |
|
1961 |
have "(\<integral>\<^isup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^isup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)" |
|
1962 |
using A by (intro positive_integral_mono_AE) auto |
|
47694 | 1963 |
then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}" |
43339 | 1964 |
using int A by (simp add: integrable_def) |
47694 | 1965 |
ultimately have "emeasure M A = 0" |
1966 |
using emeasure_nonneg[of M A] by simp |
|
1967 |
with `(emeasure M) A \<noteq> 0` show False by auto |
|
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1968 |
qed |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1969 |
ultimately show ?thesis by auto |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1970 |
qed |
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1971 |
|
43339 | 1972 |
lemma (in finite_measure) integral_less_AE_space: |
1973 |
assumes int: "integrable M X" "integrable M Y" |
|
47694 | 1974 |
assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0" |
43339 | 1975 |
shows "integral\<^isup>L M X < integral\<^isup>L M Y" |
1976 |
using gt by (intro integral_less_AE[OF int, where A="space M"]) auto |
|
1977 |
||
47694 | 1978 |
lemma integral_dominated_convergence: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1979 |
assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x" |
41705 | 1980 |
and w: "integrable M w" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1981 |
and u': "AE x in M. (\<lambda>i. u i x) ----> u' x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1982 |
and borel: "u' \<in> borel_measurable M" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1983 |
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
|
1984 |
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
|
1985 |
and "(\<lambda>i. integral\<^isup>L M (u i)) ----> integral\<^isup>L M u'" (is ?lim) |
36624 | 1986 |
proof - |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1987 |
have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1988 |
using bound by (auto simp: AE_all_countable) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1989 |
with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1990 |
by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs) |
38656 | 1991 |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1992 |
from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1993 |
by eventually_elim auto |
41705 | 1994 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1995 |
show "integrable M u'" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1996 |
by (rule integrable_bound) fact+ |
38656 | 1997 |
|
46731 | 1998 |
let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
1999 |
have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2000 |
using w u `integrable M u'` by (auto intro!: integrable_abs) |
38656 | 2001 |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2002 |
from u'_bound all_bound |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2003 |
have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2004 |
proof (eventually_elim, intro allI) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2005 |
fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2006 |
then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto |
38656 | 2007 |
also have "\<dots> \<le> w x + w x" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2008 |
using * by (intro add_mono) auto |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2009 |
finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2010 |
qed |
38656 | 2011 |
|
43920 | 2012 |
have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. ereal (?diff n x) \<partial>M) = |
2013 |
(\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" |
|
41705 | 2014 |
using diff w diff_less_2w w_pos |
38656 | 2015 |
by (subst positive_integral_diff[symmetric]) |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2016 |
(auto simp: integrable_def intro!: positive_integral_cong_AE) |
38656 | 2017 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2018 |
have "integrable M (\<lambda>x. 2 * w x)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2019 |
using w by auto |
43920 | 2020 |
hence I2w_fin: "(\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and |
2021 |
borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M" |
|
38656 | 2022 |
unfolding integrable_def by auto |
2023 |
||
43920 | 2024 |
have "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0") |
38656 | 2025 |
proof cases |
43920 | 2026 |
assume eq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0") |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2027 |
{ fix n |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2028 |
have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _") |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2029 |
using diff_less_2w unfolding positive_integral_max_0 |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2030 |
by (intro positive_integral_mono_AE) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2031 |
then have "?f n = 0" |
47694 | 2032 |
using positive_integral_positive[of M ?f'] eq_0 by auto } |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2033 |
then show ?thesis by (simp add: Limsup_const) |
38656 | 2034 |
next |
43920 | 2035 |
assume neq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0") |
2036 |
have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const) |
|
2037 |
also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2038 |
by (intro limsup_mono positive_integral_positive) |
43920 | 2039 |
finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" . |
2040 |
have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)" |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2041 |
using u' |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2042 |
proof (intro positive_integral_cong_AE, eventually_elim) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2043 |
fix x assume u': "(\<lambda>i. u i x) ----> u' x" |
43920 | 2044 |
show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))" |
2045 |
unfolding ereal_max_0 |
|
2046 |
proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal) |
|
38656 | 2047 |
have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2048 |
using u' by (safe intro!: tendsto_intros) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2049 |
then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2050 |
by (auto intro!: tendsto_real_max) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2051 |
qed (rule trivial_limit_sequentially) |
38656 | 2052 |
qed |
43920 | 2053 |
also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (ereal (?diff n x)) \<partial>M)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2054 |
using 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
|
2055 |
by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF) |
43920 | 2056 |
also have "\<dots> = (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) - |
2057 |
limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2058 |
unfolding PI_diff positive_integral_max_0 |
47694 | 2059 |
using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] |
43920 | 2060 |
by (subst liminf_ereal_cminus) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2061 |
finally show ?thesis |
47694 | 2062 |
using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2063 |
unfolding positive_integral_max_0 |
43920 | 2064 |
by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2065 |
auto |
38656 | 2066 |
qed |
41705 | 2067 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2068 |
have "liminf ?f \<le> limsup ?f" |
43920 | 2069 |
by (intro ereal_Liminf_le_Limsup trivial_limit_sequentially) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2070 |
moreover |
43920 | 2071 |
{ have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2072 |
also have "\<dots> \<le> liminf ?f" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2073 |
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
|
2074 |
finally have "0 \<le> liminf ?f" . } |
43920 | 2075 |
ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2076 |
using `limsup ?f = 0` by auto |
43920 | 2077 |
have "\<And>n. (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" |
47694 | 2078 |
using diff positive_integral_positive[of M] |
2079 |
by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
2080 |
then show ?lim_diff |
43920 | 2081 |
using ereal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq] |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2082 |
by simp |
38656 | 2083 |
|
2084 |
show ?lim |
|
2085 |
proof (rule LIMSEQ_I) |
|
2086 |
fix r :: real assume "0 < r" |
|
2087 |
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
|
2088 |
obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r" |
38656 | 2089 |
using diff by (auto simp: integral_positive) |
2090 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2091 |
show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" |
38656 | 2092 |
proof (safe intro!: exI[of _ N]) |
2093 |
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
|
2094 |
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>" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2095 |
using u `integrable M u'` by auto |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2096 |
also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'` |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2097 |
by (rule_tac integral_triangle_inequality) auto |
38656 | 2098 |
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
|
2099 |
finally show "norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" by simp |
38656 | 2100 |
qed |
2101 |
qed |
|
2102 |
qed |
|
2103 |
||
47694 | 2104 |
lemma 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
|
2105 |
assumes borel: "\<And>i. integrable M (f i)" |
38656 | 2106 |
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
|
2107 |
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
|
2108 |
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
|
2109 |
and "(\<lambda>i. integral\<^isup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral) |
38656 | 2110 |
proof - |
2111 |
have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w" |
|
2112 |
using summable unfolding summable_def by auto |
|
2113 |
from bchoice[OF this] |
|
2114 |
obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2115 |
then have w_borel: "w \<in> borel_measurable M" unfolding sums_def |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2116 |
by (rule borel_measurable_LIMSEQ) (auto simp: borel[THEN integrableD(1)]) |
38656 | 2117 |
|
46731 | 2118 |
let ?w = "\<lambda>y. if y \<in> space M then w y else 0" |
38656 | 2119 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2120 |
obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x" |
38656 | 2121 |
using sums unfolding summable_def .. |
2122 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2123 |
have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2124 |
using borel by auto |
38656 | 2125 |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2126 |
have 2: "\<And>j. AE x in M. \<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2127 |
using AE_space |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2128 |
proof eventually_elim |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2129 |
fix j x assume [simp]: "x \<in> space M" |
38656 | 2130 |
have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs) |
2131 |
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 |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2132 |
finally show "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2133 |
qed |
38656 | 2134 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2135 |
have 3: "integrable M ?w" |
38656 | 2136 |
proof (rule integral_monotone_convergence(1)) |
46731 | 2137 |
let ?F = "\<lambda>n y. (\<Sum>i = 0..<n. \<bar>f i y\<bar>)" |
2138 |
let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2139 |
have "\<And>n. integrable M (?F n)" |
38656 | 2140 |
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
|
2141 |
thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong) |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2142 |
show "AE x in M. mono (\<lambda>n. ?w' n x)" |
38656 | 2143 |
by (auto simp: mono_def le_fun_def intro!: setsum_mono2) |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2144 |
show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2145 |
using w by (simp_all add: tendsto_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
|
2146 |
have *: "\<And>n. integral\<^isup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))" |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2147 |
using borel by (simp add: integrable_abs cong: integral_cong) |
38656 | 2148 |
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
|
2149 |
show "(\<lambda>i. integral\<^isup>L M (?w' i)) ----> x" unfolding * sums_def . |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2150 |
qed (simp add: w_borel measurable_If_set) |
38656 | 2151 |
|
2152 |
from summable[THEN summable_rabs_cancel] |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2153 |
have 4: "AE x in M. (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)" |
38656 | 2154 |
by (auto intro: summable_sumr_LIMSEQ_suminf) |
2155 |
||
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2156 |
note int = integral_dominated_convergence(1,3)[OF 1 2 3 4 |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2157 |
borel_measurable_suminf[OF integrableD(1)[OF borel]]] |
38656 | 2158 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2159 |
from int show "integrable M ?S" by simp |
38656 | 2160 |
|
2161 |
show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel] |
|
2162 |
using int(2) by simp |
|
36624 | 2163 |
qed |
2164 |
||
35748 | 2165 |
section "Lebesgue integration on countable spaces" |
2166 |
||
47694 | 2167 |
lemma integral_on_countable: |
38656 | 2168 |
assumes f: "f \<in> borel_measurable M" |
35748 | 2169 |
and bij: "bij_betw enum S (f ` space M)" |
2170 |
and enum_zero: "enum ` (-S) \<subseteq> {0}" |
|
47694 | 2171 |
and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>" |
2172 |
and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)" |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2173 |
shows "integrable M f" |
47694 | 2174 |
and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums) |
35748 | 2175 |
proof - |
46731 | 2176 |
let ?A = "\<lambda>r. f -` {enum r} \<inter> space M" |
2177 |
let ?F = "\<lambda>r x. enum r * indicator (?A r) x" |
|
47694 | 2178 |
have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^isup>L M (?F r)" |
38656 | 2179 |
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator) |
35748 | 2180 |
|
38656 | 2181 |
{ fix x assume "x \<in> space M" |
2182 |
hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto |
|
2183 |
then obtain i where "i\<in>S" "enum i = f x" by auto |
|
2184 |
have F: "\<And>j. ?F j x = (if j = i then f x else 0)" |
|
2185 |
proof cases |
|
2186 |
fix j assume "j = i" |
|
2187 |
thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def) |
|
2188 |
next |
|
2189 |
fix j assume "j \<noteq> i" |
|
2190 |
show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero |
|
2191 |
by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def) |
|
2192 |
qed |
|
2193 |
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 |
|
2194 |
have "(\<lambda>i. ?F i x) sums f x" |
|
2195 |
"(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>" |
|
2196 |
by (auto intro!: sums_single simp: F F_abs) } |
|
2197 |
note F_sums_f = this(1) and F_abs_sums_f = this(2) |
|
35748 | 2198 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2199 |
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 | 2200 |
using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff) |
35748 | 2201 |
|
2202 |
{ 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
|
2203 |
have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)" |
38656 | 2204 |
by (auto simp: indicator_def intro!: integral_cong) |
47694 | 2205 |
also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))" |
38656 | 2206 |
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator) |
47694 | 2207 |
finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>" |
43920 | 2208 |
using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) } |
38656 | 2209 |
note int_abs_F = this |
35748 | 2210 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
2211 |
have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)" |
38656 | 2212 |
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator) |
2213 |
||
2214 |
have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)" |
|
2215 |
using F_abs_sums_f unfolding sums_iff by auto |
|
2216 |
||
2217 |
from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable] |
|
2218 |
show ?sums unfolding enum_eq int_f by simp |
|
2219 |
||
2220 |
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
|
2221 |
show "integrable M f" unfolding int_f by simp |
35748 | 2222 |
qed |
2223 |
||
47694 | 2224 |
section {* Distributions *} |
2225 |
||
2226 |
lemma simple_function_distr[simp]: |
|
2227 |
"simple_function (distr M M' T) f \<longleftrightarrow> simple_function M' (\<lambda>x. f x)" |
|
2228 |
unfolding simple_function_def by simp |
|
2229 |
||
2230 |
lemma positive_integral_distr: |
|
2231 |
assumes T: "T \<in> measurable M M'" |
|
2232 |
and f: "f \<in> borel_measurable M'" |
|
2233 |
shows "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)" |
|
2234 |
proof - |
|
2235 |
from borel_measurable_implies_simple_function_sequence'[OF f] |
|
2236 |
guess f' . note f' = this |
|
2237 |
then have f_distr: "\<And>i. simple_function (distr M M' T) (f' i)" |
|
2238 |
by simp |
|
2239 |
let ?f = "\<lambda>i x. f' i (T x)" |
|
2240 |
have inc: "incseq ?f" using f' by (force simp: le_fun_def incseq_def) |
|
2241 |
have sup: "\<And>x. (SUP i. ?f i x) = max 0 (f (T x))" |
|
2242 |
using f'(4) . |
|
2243 |
have sf: "\<And>i. simple_function M (\<lambda>x. f' i (T x))" |
|
2244 |
using simple_function_comp[OF T(1) f'(1)] . |
|
2245 |
show "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)" |
|
2246 |
using |
|
2247 |
positive_integral_monotone_convergence_simple[OF f'(2,5) f_distr] |
|
2248 |
positive_integral_monotone_convergence_simple[OF inc f'(5) sf] |
|
2249 |
by (simp add: positive_integral_max_0 simple_integral_distr[OF T f'(1)] f') |
|
2250 |
qed |
|
2251 |
||
2252 |
lemma integral_distr: |
|
2253 |
assumes T: "T \<in> measurable M M'" |
|
2254 |
assumes f: "f \<in> borel_measurable M'" |
|
2255 |
shows "integral\<^isup>L (distr M M' T) f = (\<integral>x. f (T x) \<partial>M)" |
|
2256 |
proof - |
|
2257 |
from measurable_comp[OF T, of f borel] |
|
2258 |
have borel: "(\<lambda>x. ereal (f x)) \<in> borel_measurable M'" "(\<lambda>x. ereal (- f x)) \<in> borel_measurable M'" |
|
2259 |
and "(\<lambda>x. f (T x)) \<in> borel_measurable M" |
|
2260 |
using f by (auto simp: comp_def) |
|
2261 |
then show ?thesis |
|
2262 |
using f unfolding lebesgue_integral_def integrable_def |
|
2263 |
by (auto simp: borel[THEN positive_integral_distr[OF T]]) |
|
2264 |
qed |
|
35692 | 2265 |
|
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2266 |
lemma integrable_distr_eq: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2267 |
assumes T: "T \<in> measurable M M'" "f \<in> borel_measurable M'" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2268 |
shows "integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2269 |
using T measurable_comp[OF T] |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2270 |
unfolding integrable_def |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2271 |
by (subst (1 2) positive_integral_distr) (auto simp: comp_def) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2272 |
|
47694 | 2273 |
lemma integrable_distr: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2274 |
assumes T: "T \<in> measurable M M'" shows "integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2275 |
by (subst integrable_distr_eq[symmetric, OF T]) auto |
47694 | 2276 |
|
2277 |
section {* Lebesgue integration on @{const count_space} *} |
|
2278 |
||
2279 |
lemma simple_function_count_space[simp]: |
|
2280 |
"simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)" |
|
2281 |
unfolding simple_function_def by simp |
|
2282 |
||
2283 |
lemma positive_integral_count_space: |
|
2284 |
assumes A: "finite {a\<in>A. 0 < f a}" |
|
2285 |
shows "integral\<^isup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
|
35582 | 2286 |
proof - |
47694 | 2287 |
have *: "(\<integral>\<^isup>+x. max 0 (f x) \<partial>count_space A) = |
2288 |
(\<integral>\<^isup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)" |
|
2289 |
by (auto intro!: positive_integral_cong |
|
2290 |
simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less) |
|
2291 |
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^isup>+ x. f a * indicator {a} x \<partial>count_space A)" |
|
2292 |
by (subst positive_integral_setsum) |
|
2293 |
(simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le) |
|
2294 |
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
|
2295 |
by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric]) |
|
2296 |
finally show ?thesis by (simp add: positive_integral_max_0) |
|
2297 |
qed |
|
2298 |
||
2299 |
lemma integrable_count_space: |
|
2300 |
"finite X \<Longrightarrow> integrable (count_space X) f" |
|
2301 |
by (auto simp: positive_integral_count_space integrable_def) |
|
2302 |
||
2303 |
lemma positive_integral_count_space_finite: |
|
2304 |
"finite A \<Longrightarrow> (\<integral>\<^isup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))" |
|
2305 |
by (subst positive_integral_max_0[symmetric]) |
|
2306 |
(auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le) |
|
2307 |
||
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2308 |
lemma lebesgue_integral_count_space_finite_support: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2309 |
assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2310 |
proof - |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2311 |
have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2312 |
"\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2313 |
"\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2314 |
"{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2315 |
"({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2316 |
by (auto split: split_max) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2317 |
have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2318 |
by (auto intro: finite_subset[OF _ f]) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2319 |
then show ?thesis |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2320 |
unfolding lebesgue_integral_def |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2321 |
apply (subst (1 2) positive_integral_max_0[symmetric]) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2322 |
apply (subst (1 2) positive_integral_count_space) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2323 |
apply (auto simp add: * setsum_negf setsum_Un |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2324 |
simp del: ereal_max) |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2325 |
done |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2326 |
qed |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2327 |
|
47694 | 2328 |
lemma lebesgue_integral_count_space_finite: |
2329 |
"finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)" |
|
2330 |
apply (auto intro!: setsum_mono_zero_left |
|
2331 |
simp: positive_integral_count_space_finite lebesgue_integral_def) |
|
2332 |
apply (subst (1 2) setsum_real_of_ereal[symmetric]) |
|
2333 |
apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong) |
|
2334 |
done |
|
2335 |
||
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2336 |
lemma borel_measurable_count_space[simp, intro!]: |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2337 |
"f \<in> borel_measurable (count_space A)" |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2338 |
by simp |
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
2339 |
|
47694 | 2340 |
section {* Measure spaces with an associated density *} |
2341 |
||
2342 |
definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
2343 |
"density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M)" |
|
35582 | 2344 |
|
47694 | 2345 |
lemma |
2346 |
shows sets_density[simp]: "sets (density M f) = sets M" |
|
2347 |
and space_density[simp]: "space (density M f) = space M" |
|
2348 |
by (auto simp: density_def) |
|
2349 |
||
2350 |
lemma |
|
2351 |
shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'" |
|
2352 |
and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'" |
|
2353 |
and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u" |
|
2354 |
unfolding measurable_def simple_function_def by simp_all |
|
2355 |
||
2356 |
lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow> |
|
2357 |
(AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'" |
|
2358 |
unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE space_closed) |
|
2359 |
||
2360 |
lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))" |
|
2361 |
proof - |
|
2362 |
have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)" |
|
2363 |
by (auto simp: indicator_def) |
|
2364 |
then show ?thesis |
|
2365 |
unfolding density_def by (simp add: positive_integral_max_0) |
|
2366 |
qed |
|
2367 |
||
2368 |
lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))" |
|
2369 |
by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max) |
|
38656 | 2370 |
|
47694 | 2371 |
lemma emeasure_density: |
2372 |
assumes f: "f \<in> borel_measurable M" and A: "A \<in> sets M" |
|
2373 |
shows "emeasure (density M f) A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)" |
|
2374 |
(is "_ = ?\<mu> A") |
|
2375 |
unfolding density_def |
|
2376 |
proof (rule emeasure_measure_of_sigma) |
|
2377 |
show "sigma_algebra (space M) (sets M)" .. |
|
2378 |
show "positive (sets M) ?\<mu>" |
|
2379 |
using f by (auto simp: positive_def intro!: positive_integral_positive) |
|
2380 |
have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^isup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'") |
|
2381 |
apply (subst positive_integral_max_0[symmetric]) |
|
2382 |
apply (intro ext positive_integral_cong_AE AE_I2) |
|
2383 |
apply (auto simp: indicator_def) |
|
2384 |
done |
|
2385 |
show "countably_additive (sets M) ?\<mu>" |
|
2386 |
unfolding \<mu>_eq |
|
2387 |
proof (intro countably_additiveI) |
|
2388 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M" |
|
2389 |
then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M" |
|
2390 |
using f by (auto intro!: borel_measurable_ereal_times) |
|
2391 |
assume disj: "disjoint_family A" |
|
2392 |
have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)" |
|
2393 |
using f * by (simp add: positive_integral_suminf) |
|
2394 |
also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f |
|
2395 |
by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE) |
|
2396 |
also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)" |
|
2397 |
unfolding suminf_indicator[OF disj] .. |
|
2398 |
finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp |
|
2399 |
qed |
|
2400 |
qed fact |
|
38656 | 2401 |
|
47694 | 2402 |
lemma null_sets_density_iff: |
2403 |
assumes f: "f \<in> borel_measurable M" |
|
2404 |
shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" |
|
2405 |
proof - |
|
2406 |
{ assume "A \<in> sets M" |
|
2407 |
have eq: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. max 0 (f x) * indicator A x \<partial>M)" |
|
2408 |
apply (subst positive_integral_max_0[symmetric]) |
|
2409 |
apply (intro positive_integral_cong) |
|
2410 |
apply (auto simp: indicator_def) |
|
2411 |
done |
|
2412 |
have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> |
|
2413 |
emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0" |
|
2414 |
unfolding eq |
|
2415 |
using f `A \<in> sets M` |
|
2416 |
by (intro positive_integral_0_iff) auto |
|
2417 |
also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)" |
|
2418 |
using f `A \<in> sets M` |
|
2419 |
by (intro AE_iff_measurable[OF _ refl, symmetric]) |
|
2420 |
(auto intro!: sets_Collect borel_measurable_ereal_eq) |
|
2421 |
also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" |
|
2422 |
by (auto simp add: indicator_def max_def split: split_if_asm) |
|
2423 |
finally have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . } |
|
2424 |
with f show ?thesis |
|
2425 |
by (simp add: null_sets_def emeasure_density cong: conj_cong) |
|
2426 |
qed |
|
2427 |
||
2428 |
lemma AE_density: |
|
2429 |
assumes f: "f \<in> borel_measurable M" |
|
2430 |
shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)" |
|
2431 |
proof |
|
2432 |
assume "AE x in density M f. P x" |
|
2433 |
with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0" |
|
2434 |
by (auto simp: eventually_ae_filter null_sets_density_iff) |
|
2435 |
then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto |
|
2436 |
with ae show "AE x in M. 0 < f x \<longrightarrow> P x" |
|
2437 |
by (rule eventually_elim2) auto |
|
2438 |
next |
|
2439 |
fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x" |
|
2440 |
then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M" |
|
2441 |
by (auto simp: eventually_ae_filter) |
|
2442 |
then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}" |
|
2443 |
"N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N" |
|
2444 |
using f by (auto simp: subset_eq intro!: sets_Collect_neg AE_not_in) |
|
2445 |
show "AE x in density M f. P x" |
|
2446 |
using ae2 |
|
2447 |
unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f] |
|
2448 |
by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *) |
|
2449 |
(auto elim: eventually_elim2) |
|
35582 | 2450 |
qed |
2451 |
||
47694 | 2452 |
lemma positive_integral_density: |
2453 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
2454 |
assumes g': "g' \<in> borel_measurable M" |
|
2455 |
shows "integral\<^isup>P (density M f) g' = (\<integral>\<^isup>+ x. f x * g' x \<partial>M)" |
|
2456 |
proof - |
|
2457 |
def g \<equiv> "\<lambda>x. max 0 (g' x)" |
|
2458 |
then have g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
2459 |
using g' by auto |
|
2460 |
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess G . note G = this |
|
2461 |
note G' = borel_measurable_simple_function[OF this(1)] simple_functionD[OF G(1)] |
|
2462 |
note G'(2)[simp] |
|
2463 |
with G(4) f g have G_M': "AE x in density M f. (SUP i. G i x) = g x" |
|
2464 |
by (auto simp add: AE_density[OF f(1)] max_def) |
|
2465 |
{ fix i |
|
2466 |
let ?I = "\<lambda>y x. indicator (G i -` {y} \<inter> space M) x" |
|
2467 |
{ fix x assume *: "x \<in> space M" "0 \<le> f x" "0 \<le> g x" |
|
2468 |
then have [simp]: "G i ` space M \<inter> {y. G i x = y \<and> x \<in> space M} = {G i x}" by auto |
|
2469 |
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))" |
|
2470 |
by (subst setsum_ereal_right_distrib) (auto simp: ac_simps) |
|
2471 |
also have "\<dots> = f x * G i x" |
|
2472 |
by (simp add: indicator_def if_distrib setsum_cases) |
|
2473 |
finally have "(\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) = f x * G i x" . } |
|
2474 |
note to_singleton = this |
|
2475 |
have "integral\<^isup>P (density M f) (G i) = integral\<^isup>S (density M f) (G i)" |
|
2476 |
using G by (intro positive_integral_eq_simple_integral) simp_all |
|
2477 |
also have "\<dots> = (\<Sum>y\<in>G i`space M. y * (\<integral>\<^isup>+x. f x * ?I y x \<partial>M))" |
|
2478 |
using f G(1) |
|
2479 |
by (auto intro!: setsum_cong arg_cong2[where f="op *"] emeasure_density |
|
2480 |
simp: simple_function_def simple_integral_def) |
|
2481 |
also have "\<dots> = (\<Sum>y\<in>G i`space M. (\<integral>\<^isup>+x. y * (f x * ?I y x) \<partial>M))" |
|
2482 |
using f G' G by (auto intro!: setsum_cong positive_integral_cmult[symmetric]) |
|
2483 |
also have "\<dots> = (\<integral>\<^isup>+x. (\<Sum>y\<in>G i`space M. y * (f x * ?I y x)) \<partial>M)" |
|
2484 |
using f G' G by (auto intro!: positive_integral_setsum[symmetric]) |
|
2485 |
finally have "integral\<^isup>P (density M f) (G i) = (\<integral>\<^isup>+x. f x * G i x \<partial>M)" |
|
2486 |
using f g G' to_singleton by (auto intro!: positive_integral_cong_AE) } |
|
2487 |
note [simp] = this |
|
2488 |
have "integral\<^isup>P (density M f) g = (SUP i. integral\<^isup>P (density M f) (G i))" using G'(1) G_M'(1) G |
|
2489 |
using positive_integral_monotone_convergence_SUP[symmetric, OF `incseq G`, of "density M f"] |
|
2490 |
by (simp cong: positive_integral_cong_AE) |
|
2491 |
also have "\<dots> = (SUP i. (\<integral>\<^isup>+x. f x * G i x \<partial>M))" by simp |
|
2492 |
also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. f x * G i x) \<partial>M)" |
|
2493 |
using f G' G(2)[THEN incseq_SucD] G |
|
2494 |
by (intro positive_integral_monotone_convergence_SUP_AE[symmetric]) |
|
2495 |
(auto simp: ereal_mult_left_mono le_fun_def ereal_zero_le_0_iff) |
|
2496 |
also have "\<dots> = (\<integral>\<^isup>+x. f x * g x \<partial>M)" using f G' G g |
|
2497 |
by (intro positive_integral_cong_AE) |
|
2498 |
(auto simp add: SUPR_ereal_cmult split: split_max) |
|
2499 |
also have "\<dots> = (\<integral>\<^isup>+x. f x * g' x \<partial>M)" |
|
2500 |
using f(2) |
|
2501 |
by (subst (2) positive_integral_max_0[symmetric]) |
|
2502 |
(auto simp: g_def max_def ereal_zero_le_0_iff intro!: positive_integral_cong_AE) |
|
2503 |
finally show "integral\<^isup>P (density M f) g' = (\<integral>\<^isup>+x. f x * g' x \<partial>M)" |
|
2504 |
unfolding g_def positive_integral_max_0 . |
|
2505 |
qed |
|
38705 | 2506 |
|
47694 | 2507 |
lemma integral_density: |
2508 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
2509 |
and g: "g \<in> borel_measurable M" |
|
2510 |
shows "integral\<^isup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)" |
|
2511 |
and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)" |
|
2512 |
unfolding lebesgue_integral_def integrable_def using f g |
|
2513 |
by (auto simp: positive_integral_density) |
|
2514 |
||
2515 |
lemma emeasure_restricted: |
|
2516 |
assumes S: "S \<in> sets M" and X: "X \<in> sets M" |
|
2517 |
shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)" |
|
38705 | 2518 |
proof - |
47694 | 2519 |
have "emeasure (density M (indicator S)) X = (\<integral>\<^isup>+x. indicator S x * indicator X x \<partial>M)" |
2520 |
using S X by (simp add: emeasure_density) |
|
2521 |
also have "\<dots> = (\<integral>\<^isup>+x. indicator (S \<inter> X) x \<partial>M)" |
|
2522 |
by (auto intro!: positive_integral_cong simp: indicator_def) |
|
2523 |
also have "\<dots> = emeasure M (S \<inter> X)" |
|
2524 |
using S X by (simp add: Int) |
|
2525 |
finally show ?thesis . |
|
2526 |
qed |
|
2527 |
||
2528 |
lemma measure_restricted: |
|
2529 |
"S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)" |
|
2530 |
by (simp add: emeasure_restricted measure_def) |
|
2531 |
||
2532 |
lemma (in finite_measure) finite_measure_restricted: |
|
2533 |
"S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))" |
|
2534 |
by default (simp add: emeasure_restricted) |
|
2535 |
||
2536 |
lemma emeasure_density_const: |
|
2537 |
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A" |
|
2538 |
by (auto simp: positive_integral_cmult_indicator emeasure_density) |
|
2539 |
||
2540 |
lemma measure_density_const: |
|
2541 |
"A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A" |
|
2542 |
by (auto simp: emeasure_density_const measure_def) |
|
2543 |
||
2544 |
lemma density_density_eq: |
|
2545 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> |
|
2546 |
density (density M f) g = density M (\<lambda>x. f x * g x)" |
|
2547 |
by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps) |
|
2548 |
||
2549 |
lemma distr_density_distr: |
|
2550 |
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
2551 |
and inv: "\<forall>x\<in>space M. T' (T x) = x" |
|
2552 |
assumes f: "f \<in> borel_measurable M'" |
|
2553 |
shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L") |
|
2554 |
proof (rule measure_eqI) |
|
2555 |
fix A assume A: "A \<in> sets ?R" |
|
2556 |
{ fix x assume "x \<in> space M" |
|
2557 |
with sets_into_space[OF A] |
|
2558 |
have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)" |
|
2559 |
using T inv by (auto simp: indicator_def measurable_space) } |
|
2560 |
with A T T' f show "emeasure ?R A = emeasure ?L A" |
|
2561 |
by (simp add: measurable_comp emeasure_density emeasure_distr |
|
2562 |
positive_integral_distr measurable_sets cong: positive_integral_cong) |
|
2563 |
qed simp |
|
2564 |
||
2565 |
lemma density_density_divide: |
|
2566 |
fixes f g :: "'a \<Rightarrow> real" |
|
2567 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
2568 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
2569 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
2570 |
shows "density (density M f) (\<lambda>x. g x / f x) = density M g" |
|
2571 |
proof - |
|
2572 |
have "density M g = density M (\<lambda>x. f x * (g x / f x))" |
|
2573 |
using f g ac by (auto intro!: density_cong measurable_If) |
|
2574 |
then show ?thesis |
|
2575 |
using f g by (subst density_density_eq) auto |
|
38705 | 2576 |
qed |
2577 |
||
47694 | 2578 |
section {* Point measure *} |
2579 |
||
2580 |
definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
2581 |
"point_measure A f = density (count_space A) f" |
|
2582 |
||
2583 |
lemma |
|
2584 |
shows space_point_measure: "space (point_measure A f) = A" |
|
2585 |
and sets_point_measure: "sets (point_measure A f) = Pow A" |
|
2586 |
by (auto simp: point_measure_def) |
|
2587 |
||
2588 |
lemma measurable_point_measure_eq1[simp]: |
|
2589 |
"g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M" |
|
2590 |
unfolding point_measure_def by simp |
|
2591 |
||
2592 |
lemma measurable_point_measure_eq2_finite[simp]: |
|
2593 |
"finite A \<Longrightarrow> |
|
2594 |
g \<in> measurable M (point_measure A f) \<longleftrightarrow> |
|
2595 |
(g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))" |
|
2596 |
unfolding point_measure_def by simp |
|
2597 |
||
2598 |
lemma simple_function_point_measure[simp]: |
|
2599 |
"simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)" |
|
2600 |
by (simp add: point_measure_def) |
|
2601 |
||
2602 |
lemma emeasure_point_measure: |
|
2603 |
assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A" |
|
2604 |
shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)" |
|
35977 | 2605 |
proof - |
47694 | 2606 |
have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}" |
2607 |
using `X \<subseteq> A` by auto |
|
2608 |
with A show ?thesis |
|
2609 |
by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff |
|
2610 |
point_measure_def indicator_def) |
|
35977 | 2611 |
qed |
2612 |
||
47694 | 2613 |
lemma emeasure_point_measure_finite: |
49795 | 2614 |
"finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)" |
47694 | 2615 |
by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le) |
2616 |
||
49795 | 2617 |
lemma emeasure_point_measure_finite2: |
2618 |
"X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)" |
|
2619 |
by (subst emeasure_point_measure) |
|
2620 |
(auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le) |
|
2621 |
||
47694 | 2622 |
lemma null_sets_point_measure_iff: |
2623 |
"X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)" |
|
2624 |
by (auto simp: AE_count_space null_sets_density_iff point_measure_def) |
|
2625 |
||
2626 |
lemma AE_point_measure: |
|
2627 |
"(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)" |
|
2628 |
unfolding point_measure_def |
|
2629 |
by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def) |
|
2630 |
||
2631 |
lemma positive_integral_point_measure: |
|
2632 |
"finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow> |
|
2633 |
integral\<^isup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)" |
|
2634 |
unfolding point_measure_def |
|
2635 |
apply (subst density_max_0) |
|
2636 |
apply (subst positive_integral_density) |
|
2637 |
apply (simp_all add: AE_count_space positive_integral_density) |
|
2638 |
apply (subst positive_integral_count_space ) |
|
2639 |
apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff) |
|
2640 |
apply (rule finite_subset) |
|
2641 |
prefer 2 |
|
2642 |
apply assumption |
|
2643 |
apply auto |
|
2644 |
done |
|
2645 |
||
2646 |
lemma positive_integral_point_measure_finite: |
|
2647 |
"finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow> |
|
2648 |
integral\<^isup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)" |
|
2649 |
by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le) |
|
2650 |
||
2651 |
lemma lebesgue_integral_point_measure_finite: |
|
2652 |
"finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^isup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)" |
|
2653 |
by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def) |
|
2654 |
||
2655 |
lemma integrable_point_measure_finite: |
|
2656 |
"finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g" |
|
2657 |
unfolding point_measure_def |
|
2658 |
apply (subst density_ereal_max_0) |
|
2659 |
apply (subst integral_density) |
|
2660 |
apply (auto simp: AE_count_space integrable_count_space) |
|
2661 |
done |
|
2662 |
||
2663 |
section {* Uniform measure *} |
|
2664 |
||
2665 |
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)" |
|
2666 |
||
2667 |
lemma |
|
2668 |
shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M" |
|
2669 |
and space_uniform_measure[simp]: "space (uniform_measure M A) = space M" |
|
2670 |
by (auto simp: uniform_measure_def) |
|
2671 |
||
2672 |
lemma emeasure_uniform_measure[simp]: |
|
2673 |
assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
|
2674 |
shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A" |
|
2675 |
proof - |
|
2676 |
from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^isup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)" |
|
2677 |
by (auto simp add: uniform_measure_def emeasure_density split: split_indicator |
|
2678 |
intro!: positive_integral_cong) |
|
2679 |
also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A" |
|
2680 |
using A B |
|
2681 |
by (subst positive_integral_cmult_indicator) (simp_all add: Int emeasure_nonneg) |
|
2682 |
finally show ?thesis . |
|
2683 |
qed |
|
2684 |
||
2685 |
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M" |
|
2686 |
using emeasure_notin_sets[of A M] by blast |
|
2687 |
||
2688 |
lemma measure_uniform_measure[simp]: |
|
2689 |
assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M" |
|
2690 |
shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A" |
|
2691 |
using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A |
|
2692 |
by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def) |
|
2693 |
||
2694 |
section {* Uniform count measure *} |
|
2695 |
||
2696 |
definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)" |
|
2697 |
||
2698 |
lemma |
|
2699 |
shows space_uniform_count_measure: "space (uniform_count_measure A) = A" |
|
2700 |
and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A" |
|
2701 |
unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure) |
|
2702 |
||
2703 |
lemma emeasure_uniform_count_measure: |
|
2704 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A" |
|
2705 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def) |
|
2706 |
||
2707 |
lemma measure_uniform_count_measure: |
|
2708 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A" |
|
2709 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def) |
|
2710 |
||
35748 | 2711 |
end |