author | Andreas Lochbihler |
Tue, 14 Apr 2015 14:12:19 +0200 | |
changeset 60064 | 63124d48a2ee |
parent 59779 | b6bda9140e39 |
child 60175 | 831ddb69db9b |
permissions | -rw-r--r-- |
56993
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introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
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1 |
(* Title: HOL/Probability/Nonnegative_Lebesgue_Integration.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
Author: Armin Heller, TU München |
|
4 |
*) |
|
38656 | 5 |
|
58876 | 6 |
section {* Lebesgue Integration for Nonnegative Functions *} |
35582 | 7 |
|
56993
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introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
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8 |
theory Nonnegative_Lebesgue_Integration |
47694 | 9 |
imports Measure_Space Borel_Space |
35582 | 10 |
begin |
11 |
||
59426
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integral of the product of count spaces equals the integral of the count space of the product type
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|
12 |
lemma infinite_countable_subset': |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
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|
13 |
assumes X: "infinite X" shows "\<exists>C\<subseteq>X. countable C \<and> infinite C" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
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14 |
proof - |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
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|
15 |
from infinite_countable_subset[OF X] guess f .. |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
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|
16 |
then show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
17 |
by (intro exI[of _ "range f"]) (auto simp: range_inj_infinite) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
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|
18 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
19 |
|
56949 | 20 |
lemma indicator_less_ereal[simp]: |
21 |
"indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)" |
|
22 |
by (simp add: indicator_def not_le) |
|
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reworked Probability theory: measures are not type restricted to positive extended reals
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23 |
|
56994 | 24 |
subsection "Simple function" |
35582 | 25 |
|
38656 | 26 |
text {* |
27 |
||
56996 | 28 |
Our simple functions are not restricted to nonnegative real numbers. Instead |
38656 | 29 |
they are just functions with a finite range and are measurable when singleton |
30 |
sets are measurable. |
|
35582 | 31 |
|
38656 | 32 |
*} |
33 |
||
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
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parents:
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34 |
definition "simple_function M g \<longleftrightarrow> |
38656 | 35 |
finite (g ` space M) \<and> |
36 |
(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)" |
|
36624 | 37 |
|
47694 | 38 |
lemma simple_functionD: |
41689
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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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|
39 |
assumes "simple_function M g" |
40875 | 40 |
shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M" |
40871 | 41 |
proof - |
42 |
show "finite (g ` space M)" |
|
43 |
using assms unfolding simple_function_def by auto |
|
40875 | 44 |
have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto |
45 |
also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto |
|
46 |
finally show "g -` X \<inter> space M \<in> sets M" using assms |
|
50002
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add measurability prover; add support for Borel sets
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by (auto simp del: UN_simps simp: simple_function_def) |
40871 | 48 |
qed |
36624 | 49 |
|
56949 | 50 |
lemma measurable_simple_function[measurable_dest]: |
51 |
"simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)" |
|
52 |
unfolding simple_function_def measurable_def |
|
53 |
proof safe |
|
54 |
fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M" |
|
55 |
then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M" |
|
56 |
by (intro sets.finite_UN) auto |
|
57 |
also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M" |
|
58 |
by (auto split: split_if_asm) |
|
59 |
finally show "f -` A \<inter> space M \<in> sets M" . |
|
60 |
qed simp |
|
61 |
||
62 |
lemma borel_measurable_simple_function: |
|
63 |
"simple_function M f \<Longrightarrow> f \<in> borel_measurable M" |
|
64 |
by (auto dest!: measurable_simple_function simp: measurable_def) |
|
65 |
||
47694 | 66 |
lemma simple_function_measurable2[intro]: |
41981
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reworked Probability theory: measures are not type restricted to positive extended reals
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67 |
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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|
68 |
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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|
69 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
70 |
have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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|
71 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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|
72 |
then show ?thesis using assms[THEN simple_functionD(2)] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
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73 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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74 |
|
47694 | 75 |
lemma simple_function_indicator_representation: |
43920 | 76 |
fixes f ::"'a \<Rightarrow> ereal" |
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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diff
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|
77 |
assumes f: "simple_function M f" and x: "x \<in> space M" |
38656 | 78 |
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)" |
79 |
(is "?l = ?r") |
|
80 |
proof - |
|
38705 | 81 |
have "?r = (\<Sum>y \<in> f ` space M. |
38656 | 82 |
(if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))" |
57418 | 83 |
by (auto intro!: setsum.cong) |
38656 | 84 |
also have "... = f x * indicator (f -` {f x} \<inter> space M) x" |
57418 | 85 |
using assms by (auto dest: simple_functionD simp: setsum.delta) |
38656 | 86 |
also have "... = f x" using x by (auto simp: indicator_def) |
87 |
finally show ?thesis by auto |
|
88 |
qed |
|
36624 | 89 |
|
47694 | 90 |
lemma simple_function_notspace: |
43920 | 91 |
"simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h") |
35692 | 92 |
proof - |
38656 | 93 |
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto |
94 |
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset) |
|
95 |
have "?h -` {0} \<inter> space M = space M" by auto |
|
96 |
thus ?thesis unfolding simple_function_def by auto |
|
97 |
qed |
|
98 |
||
47694 | 99 |
lemma simple_function_cong: |
38656 | 100 |
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:
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|
101 |
shows "simple_function M f \<longleftrightarrow> simple_function M g" |
38656 | 102 |
proof - |
103 |
have "f ` space M = g ` space M" |
|
104 |
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
105 |
using assms by (auto intro!: image_eqI) |
|
106 |
thus ?thesis unfolding simple_function_def using assms by simp |
|
107 |
qed |
|
108 |
||
47694 | 109 |
lemma simple_function_cong_algebra: |
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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|
110 |
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
|
111 |
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
|
112 |
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
|
113 |
|
47694 | 114 |
lemma simple_function_borel_measurable: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
115 |
fixes f :: "'a \<Rightarrow> 'x::{t2_space}" |
38656 | 116 |
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
|
117 |
shows "simple_function M f" |
38656 | 118 |
using assms unfolding simple_function_def |
119 |
by (auto intro: borel_measurable_vimage) |
|
120 |
||
56949 | 121 |
lemma simple_function_eq_measurable: |
43920 | 122 |
fixes f :: "'a \<Rightarrow> ereal" |
56949 | 123 |
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)" |
124 |
using simple_function_borel_measurable[of f] measurable_simple_function[of M f] |
|
44890
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new fastforce replacing fastsimp - less confusing name
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diff
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|
125 |
by (fastforce simp: simple_function_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
126 |
|
47694 | 127 |
lemma simple_function_const[intro, simp]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
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|
128 |
"simple_function M (\<lambda>x. c)" |
38656 | 129 |
by (auto intro: finite_subset simp: simple_function_def) |
47694 | 130 |
lemma simple_function_compose[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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diff
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|
131 |
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
|
132 |
shows "simple_function M (g \<circ> f)" |
38656 | 133 |
unfolding simple_function_def |
134 |
proof safe |
|
135 |
show "finite ((g \<circ> f) ` space M)" |
|
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
54611
diff
changeset
|
136 |
using assms unfolding simple_function_def by (auto simp: image_comp [symmetric]) |
38656 | 137 |
next |
138 |
fix x assume "x \<in> space M" |
|
139 |
let ?G = "g -` {g (f x)} \<inter> (f`space M)" |
|
140 |
have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M = |
|
141 |
(\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto |
|
142 |
show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M" |
|
143 |
using assms unfolding simple_function_def * |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
144 |
by (rule_tac sets.finite_UN) auto |
38656 | 145 |
qed |
146 |
||
47694 | 147 |
lemma simple_function_indicator[intro, simp]: |
38656 | 148 |
assumes "A \<in> sets M" |
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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|
149 |
shows "simple_function M (indicator A)" |
35692 | 150 |
proof - |
38656 | 151 |
have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _") |
152 |
by (auto simp: indicator_def) |
|
153 |
hence "finite ?S" by (rule finite_subset) simp |
|
154 |
moreover have "- A \<inter> space M = space M - A" by auto |
|
155 |
ultimately show ?thesis unfolding simple_function_def |
|
46905 | 156 |
using assms by (auto simp: indicator_def [abs_def]) |
35692 | 157 |
qed |
158 |
||
47694 | 159 |
lemma simple_function_Pair[intro, simp]: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
160 |
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
|
161 |
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
|
162 |
shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p") |
38656 | 163 |
unfolding simple_function_def |
164 |
proof safe |
|
165 |
show "finite (?p ` space M)" |
|
166 |
using assms unfolding simple_function_def |
|
167 |
by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto |
|
168 |
next |
|
169 |
fix x assume "x \<in> space M" |
|
170 |
have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = |
|
171 |
(f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" |
|
172 |
by auto |
|
173 |
with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M" |
|
174 |
using assms unfolding simple_function_def by auto |
|
175 |
qed |
|
35692 | 176 |
|
47694 | 177 |
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
|
178 |
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
|
179 |
shows "simple_function M (\<lambda>x. g (f x))" |
38656 | 180 |
using simple_function_compose[OF assms, of g] |
181 |
by (simp add: comp_def) |
|
35582 | 182 |
|
47694 | 183 |
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
|
184 |
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
|
185 |
shows "simple_function M (\<lambda>x. h (f x) (g x))" |
38656 | 186 |
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
|
187 |
have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))" |
38656 | 188 |
using assms by auto |
189 |
thus ?thesis by (simp_all add: comp_def) |
|
190 |
qed |
|
35582 | 191 |
|
47694 | 192 |
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"] |
38656 | 193 |
and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"] |
194 |
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"] |
|
195 |
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"] |
|
196 |
and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"] |
|
197 |
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"] |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
198 |
and simple_function_max[intro, simp] = simple_function_compose2[where h=max] |
38656 | 199 |
|
47694 | 200 |
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
|
201 |
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
|
202 |
shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)" |
38656 | 203 |
proof cases |
204 |
assume "finite P" from this assms show ?thesis by induct auto |
|
205 |
qed auto |
|
35582 | 206 |
|
56949 | 207 |
lemma simple_function_ereal[intro, simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
208 |
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f" |
56949 | 209 |
shows "simple_function M (\<lambda>x. ereal (f x))" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
210 |
by (rule simple_function_compose1[OF sf]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
211 |
|
56949 | 212 |
lemma simple_function_real_of_nat[intro, simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
213 |
fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f" |
56949 | 214 |
shows "simple_function M (\<lambda>x. real (f x))" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
215 |
by (rule simple_function_compose1[OF sf]) |
35582 | 216 |
|
47694 | 217 |
lemma borel_measurable_implies_simple_function_sequence: |
43920 | 218 |
fixes u :: "'a \<Rightarrow> ereal" |
38656 | 219 |
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
|
220 |
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
|
221 |
(\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)" |
35582 | 222 |
proof - |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
223 |
def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else nat(floor (real (u x) * 2 ^ i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
224 |
{ 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
|
225 |
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
|
226 |
assume "\<not> real j \<le> u x" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
227 |
then have "nat(floor (real (u x) * 2 ^ j)) \<le> nat(floor (j * 2 ^ j))" |
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
228 |
by (cases "u x") (auto intro!: nat_mono floor_mono) |
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
229 |
moreover have "real (nat(floor (j * 2 ^ j))) \<le> j * 2^j" |
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
230 |
by linarith |
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
231 |
ultimately show "nat(floor (real (u x) * 2 ^ j)) \<le> j * 2 ^ j" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
232 |
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
|
233 |
qed auto } |
38656 | 234 |
note f_upper = this |
35582 | 235 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
236 |
have real_f: |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
237 |
"\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (nat(floor (real (u x) * 2 ^ i))))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
238 |
unfolding f_def by auto |
35582 | 239 |
|
46731 | 240 |
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
|
241 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
242 |
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
|
243 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
244 |
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
|
245 |
proof (intro simple_function_borel_measurable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
246 |
show "(\<lambda>x. real (f x i)) \<in> borel_measurable M" |
50021
d96a3f468203
add support for function application to measurability prover
hoelzl
parents:
50003
diff
changeset
|
247 |
using u by (auto simp: real_f) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
248 |
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
|
249 |
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
|
250 |
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
|
251 |
by (rule finite_subset) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
252 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
253 |
then show "simple_function M (?g i)" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
254 |
by (auto) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
255 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
256 |
show "incseq ?g" |
43920 | 257 |
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
|
258 |
fix x and i :: nat |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
259 |
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
|
260 |
proof ((split split_if)+, intro conjI impI) |
43920 | 261 |
assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
262 |
then show "i * 2 ^ i * 2 \<le> nat(floor (real (u x) * 2 ^ Suc i))" |
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
263 |
by (cases "u x") (auto intro!: le_nat_floor) |
38656 | 264 |
next |
43920 | 265 |
assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
266 |
then show "nat(floor (real (u x) * 2 ^ i)) * 2 \<le> Suc i * 2 ^ Suc i" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
267 |
by (cases "u x") auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
268 |
next |
43920 | 269 |
assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
270 |
have "nat(floor (real (u x) * 2 ^ i)) * 2 = nat(floor (real (u x) * 2 ^ i)) * nat(floor(2::real))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
271 |
by simp |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
272 |
also have "\<dots> \<le> nat(floor (real (u x) * 2 ^ i * 2))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
273 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
274 |
assume "0 \<le> u x" then show ?thesis |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
275 |
by (intro le_mult_nat_floor) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
276 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
277 |
assume "\<not> 0 \<le> u x" then show ?thesis |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
278 |
by (cases "u x") (auto simp: nat_floor_neg mult_nonpos_nonneg) |
38656 | 279 |
qed |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
280 |
also have "\<dots> = nat(floor (real (u x) * 2 ^ Suc i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
281 |
by (simp add: ac_simps) |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
282 |
finally show "nat(floor (real (u x) * 2 ^ i)) * 2 \<le> nat(floor (real (u x) * 2 ^ Suc i))" . |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
283 |
qed simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
284 |
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
|
285 |
by (auto simp: field_simps) |
35582 | 286 |
qed |
38656 | 287 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
288 |
fix x show "(SUP i. ?g i x) = max 0 (u x)" |
51000 | 289 |
proof (rule SUP_eqI) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
290 |
fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
291 |
by (cases "u x") (auto simp: field_simps nat_floor_neg mult_nonpos_nonneg) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
292 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
293 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y" |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56537
diff
changeset
|
294 |
have "\<And>i. 0 \<le> ?g i x" by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
295 |
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
|
296 |
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
|
297 |
proof (cases y) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
298 |
case (real r) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
299 |
with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq) |
44666 | 300 |
from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le) |
43920 | 301 |
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
|
302 |
then guess p .. note ux = this |
44666 | 303 |
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
|
304 |
have "p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
305 |
proof (rule ccontr) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
306 |
assume "\<not> p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
307 |
with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"] |
56536 | 308 |
obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: field_simps) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
309 |
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
|
310 |
moreover |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
311 |
have "real (nat(floor (p * 2 ^ max N m))) \<le> r * 2 ^ max N m" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
312 |
using *[of "max N m"] m unfolding real_f using ux |
56536 | 313 |
by (cases "0 \<le> u x") (simp_all add: max_def split: split_if_asm) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
314 |
then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
315 |
by linarith |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
316 |
ultimately show False by auto |
38656 | 317 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
318 |
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
|
319 |
qed (insert `0 \<le> y`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
320 |
qed |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56537
diff
changeset
|
321 |
qed auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
322 |
qed |
35582 | 323 |
|
47694 | 324 |
lemma borel_measurable_implies_simple_function_sequence': |
43920 | 325 |
fixes u :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
326 |
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
|
327 |
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
|
328 |
"\<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
|
329 |
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
|
330 |
|
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
331 |
lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]: |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
332 |
fixes u :: "'a \<Rightarrow> ereal" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
333 |
assumes u: "simple_function M u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
334 |
assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
335 |
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
336 |
assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
337 |
assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
338 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
339 |
proof (rule cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
340 |
from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
341 |
proof eventually_elim |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
342 |
fix x assume x: "x \<in> space M" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
343 |
from simple_function_indicator_representation[OF u x] |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
344 |
show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" .. |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
345 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
346 |
next |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
347 |
from u have "finite (u ` space M)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
348 |
unfolding simple_function_def by auto |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
349 |
then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
350 |
proof induct |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
351 |
case empty show ?case |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
352 |
using set[of "{}"] by (simp add: indicator_def[abs_def]) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
353 |
qed (auto intro!: add mult set simple_functionD u) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
354 |
next |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
355 |
show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
356 |
apply (subst simple_function_cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
357 |
apply (rule simple_function_indicator_representation[symmetric]) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
358 |
apply (auto intro: u) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
359 |
done |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
360 |
qed fact |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
361 |
|
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
362 |
lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]: |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
363 |
fixes u :: "'a \<Rightarrow> ereal" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
364 |
assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
365 |
assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
366 |
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)" |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
367 |
assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
368 |
assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
369 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
370 |
proof - |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
371 |
show ?thesis |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
372 |
proof (rule cong) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
373 |
fix x assume x: "x \<in> space M" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
374 |
from simple_function_indicator_representation[OF u x] |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
375 |
show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" .. |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
376 |
next |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
377 |
show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
378 |
apply (subst simple_function_cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
379 |
apply (rule simple_function_indicator_representation[symmetric]) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
380 |
apply (auto intro: u) |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
381 |
done |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
382 |
next |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
383 |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
384 |
from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
385 |
unfolding simple_function_def by auto |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
386 |
then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
387 |
proof induct |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
388 |
case empty show ?case |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
389 |
using set[of "{}"] by (simp add: indicator_def[abs_def]) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
390 |
next |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
391 |
case (insert x S) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
392 |
{ fix z have "(\<Sum>y\<in>S. y * indicator (u -` {y} \<inter> space M) z) = 0 \<or> |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
393 |
x * indicator (u -` {x} \<inter> space M) z = 0" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
394 |
using insert by (subst setsum_ereal_0) (auto simp: indicator_def) } |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
395 |
note disj = this |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
396 |
from insert show ?case |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
397 |
by (auto intro!: add mult set simple_functionD u setsum_nonneg simple_function_setsum disj) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
398 |
qed |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
399 |
qed fact |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
400 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
401 |
|
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
402 |
lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]: |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
403 |
fixes u :: "'a \<Rightarrow> ereal" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
404 |
assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
405 |
assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
406 |
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
407 |
assumes mult': "\<And>u c. 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
408 |
assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> v x < \<infinity>) \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
409 |
assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i x. x \<in> space M \<Longrightarrow> U i x < \<infinity>) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> u = (SUP i. U i) \<Longrightarrow> P (SUP i. U i)" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
410 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
411 |
using u |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
412 |
proof (induct rule: borel_measurable_implies_simple_function_sequence') |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
413 |
fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
414 |
sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
415 |
have u_eq: "u = (SUP i. U i)" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
416 |
using nn u sup by (auto simp: max_def) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
417 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
418 |
have not_inf: "\<And>x i. x \<in> space M \<Longrightarrow> U i x < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
419 |
using U by (auto simp: image_iff eq_commute) |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
420 |
|
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
421 |
from U have "\<And>i. U i \<in> borel_measurable M" |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
422 |
by (simp add: borel_measurable_simple_function) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
423 |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
424 |
show "P u" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
425 |
unfolding u_eq |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
426 |
proof (rule seq) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
427 |
fix i show "P (U i)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
428 |
using `simple_function M (U i)` nn[of i] not_inf[of _ i] |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
429 |
proof (induct rule: simple_function_induct_nn) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
430 |
case (mult u c) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
431 |
show ?case |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
432 |
proof cases |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
433 |
assume "c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
434 |
with mult(2) show ?thesis |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
435 |
by (intro cong[of "\<lambda>x. c * u x" "indicator {}"] set) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
436 |
(auto dest!: borel_measurable_simple_function) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
437 |
next |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
438 |
assume "\<not> (c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0))" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
439 |
with mult obtain x where u_fin: "\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
440 |
and x: "x \<in> space M" "u x \<noteq> 0" "c \<noteq> 0" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
441 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
442 |
with mult have "P u" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
443 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
444 |
from x mult(5)[OF `x \<in> space M`] mult(1) mult(3)[of x] have "c < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
445 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
446 |
with u_fin mult |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
447 |
show ?thesis |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
448 |
by (intro mult') (auto dest!: borel_measurable_simple_function) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
449 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
450 |
qed (auto intro: cong intro!: set add dest!: borel_measurable_simple_function) |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
451 |
qed fact+ |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
452 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
453 |
|
47694 | 454 |
lemma simple_function_If_set: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
455 |
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
|
456 |
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
|
457 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
458 |
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
|
459 |
show ?thesis unfolding simple_function_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
460 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
461 |
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
|
462 |
from finite_subset[OF this] assms |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
463 |
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
|
464 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
465 |
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
|
466 |
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
|
467 |
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
|
468 |
else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
469 |
using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
470 |
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
|
471 |
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
|
472 |
show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto |
35582 | 473 |
qed |
474 |
qed |
|
475 |
||
47694 | 476 |
lemma simple_function_If: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
477 |
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
|
478 |
shows "simple_function M (\<lambda>x. if P x then f x else g x)" |
35582 | 479 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
480 |
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
|
481 |
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp |
38656 | 482 |
qed |
483 |
||
47694 | 484 |
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
|
485 |
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
|
486 |
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
|
487 |
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
|
488 |
using assms unfolding simple_function_def by auto |
39092 | 489 |
|
47694 | 490 |
lemma simple_function_comp: |
491 |
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
|
492 |
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
|
493 |
shows "simple_function M (\<lambda>x. f (T x))" |
41661 | 494 |
proof (intro simple_function_def[THEN iffD2] conjI ballI) |
495 |
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'" |
|
496 |
using T unfolding measurable_def by auto |
|
497 |
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
|
498 |
using f unfolding simple_function_def by (auto intro: finite_subset) |
41661 | 499 |
fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M" |
500 |
then have "i \<in> f ` space M'" |
|
501 |
using T unfolding measurable_def by auto |
|
502 |
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
|
503 |
using f unfolding simple_function_def by auto |
41661 | 504 |
then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M" |
505 |
using T unfolding measurable_def by auto |
|
506 |
also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M" |
|
507 |
using T unfolding measurable_def by auto |
|
508 |
finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" . |
|
40859 | 509 |
qed |
510 |
||
56994 | 511 |
subsection "Simple integral" |
38656 | 512 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
513 |
definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
514 |
"integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
515 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
516 |
syntax |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
517 |
"_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
518 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
519 |
translations |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
520 |
"\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)" |
35582 | 521 |
|
47694 | 522 |
lemma simple_integral_cong: |
38656 | 523 |
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
524 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
38656 | 525 |
proof - |
526 |
have "f ` space M = g ` space M" |
|
527 |
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
528 |
using assms by (auto intro!: image_eqI) |
|
529 |
thus ?thesis unfolding simple_integral_def by simp |
|
530 |
qed |
|
531 |
||
47694 | 532 |
lemma simple_integral_const[simp]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
533 |
"(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)" |
38656 | 534 |
proof (cases "space M = {}") |
535 |
case True thus ?thesis unfolding simple_integral_def by simp |
|
536 |
next |
|
537 |
case False hence "(\<lambda>x. c) ` space M = {c}" by auto |
|
538 |
thus ?thesis unfolding simple_integral_def by simp |
|
35582 | 539 |
qed |
540 |
||
47694 | 541 |
lemma simple_function_partition: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
542 |
assumes f: "simple_function M f" and g: "simple_function M g" |
56949 | 543 |
assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y" |
544 |
assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)" |
|
545 |
shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})" |
|
546 |
(is "_ = ?r") |
|
547 |
proof - |
|
548 |
from f g have [simp]: "finite (f`space M)" "finite (g`space M)" |
|
549 |
by (auto simp: simple_function_def) |
|
550 |
from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)" |
|
551 |
by (auto intro: measurable_simple_function) |
|
35582 | 552 |
|
56949 | 553 |
{ fix y assume "y \<in> space M" |
554 |
then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}" |
|
555 |
by (auto cong: sub simp: v[symmetric]) } |
|
556 |
note eq = this |
|
35582 | 557 |
|
56949 | 558 |
have "integral\<^sup>S M f = |
559 |
(\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. |
|
560 |
if \<exists>x\<in>space M. y = f x \<and> z = g x then emeasure M {x\<in>space M. g x = z} else 0))" |
|
561 |
unfolding simple_integral_def |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
562 |
proof (safe intro!: setsum.cong ereal_right_mult_cong) |
56949 | 563 |
fix y assume y: "y \<in> space M" "f y \<noteq> 0" |
564 |
have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = |
|
565 |
{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}" |
|
566 |
by auto |
|
567 |
have eq:"(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i}) = |
|
568 |
f -` {f y} \<inter> space M" |
|
569 |
by (auto simp: eq_commute cong: sub rev_conj_cong) |
|
570 |
have "finite (g`space M)" by simp |
|
571 |
then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}" |
|
572 |
by (rule rev_finite_subset) auto |
|
573 |
then show "emeasure M (f -` {f y} \<inter> space M) = |
|
574 |
(\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then emeasure M {x \<in> space M. g x = z} else 0)" |
|
57418 | 575 |
apply (simp add: setsum.If_cases) |
56949 | 576 |
apply (subst setsum_emeasure) |
577 |
apply (auto simp: disjoint_family_on_def eq) |
|
578 |
done |
|
38656 | 579 |
qed |
56949 | 580 |
also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. |
581 |
if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))" |
|
57418 | 582 |
by (auto intro!: setsum.cong simp: setsum_ereal_right_distrib emeasure_nonneg) |
56949 | 583 |
also have "\<dots> = ?r" |
57418 | 584 |
by (subst setsum.commute) |
585 |
(auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq) |
|
56949 | 586 |
finally show "integral\<^sup>S M f = ?r" . |
35582 | 587 |
qed |
588 |
||
47694 | 589 |
lemma simple_integral_add[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
590 |
assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
591 |
shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g" |
35582 | 592 |
proof - |
56949 | 593 |
have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = |
594 |
(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. (fst y + snd y) * emeasure M {x\<in>space M. (f x, g x) = y})" |
|
595 |
by (intro simple_function_partition) (auto intro: f g) |
|
596 |
also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) + |
|
597 |
(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})" |
|
57418 | 598 |
using assms(2,4) by (auto intro!: setsum.cong ereal_left_distrib simp: setsum.distrib[symmetric]) |
56949 | 599 |
also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. f x \<partial>M)" |
600 |
by (intro simple_function_partition[symmetric]) (auto intro: f g) |
|
601 |
also have "(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y}) = (\<integral>\<^sup>Sx. g x \<partial>M)" |
|
602 |
by (intro simple_function_partition[symmetric]) (auto intro: f g) |
|
603 |
finally show ?thesis . |
|
35582 | 604 |
qed |
605 |
||
47694 | 606 |
lemma simple_integral_setsum[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
607 |
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
|
608 |
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
609 |
shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))" |
38656 | 610 |
proof cases |
611 |
assume "finite P" |
|
612 |
from this assms show ?thesis |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
613 |
by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg) |
38656 | 614 |
qed auto |
615 |
||
47694 | 616 |
lemma simple_integral_mult[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
617 |
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
618 |
shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f" |
38656 | 619 |
proof - |
56949 | 620 |
have "(\<integral>\<^sup>Sx. c * f x \<partial>M) = (\<Sum>y\<in>f ` space M. (c * y) * emeasure M {x\<in>space M. f x = y})" |
621 |
using f by (intro simple_function_partition) auto |
|
622 |
also have "\<dots> = c * integral\<^sup>S M f" |
|
623 |
using f unfolding simple_integral_def |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
624 |
by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult.assoc Int_def conj_commute) |
56949 | 625 |
finally show ?thesis . |
40871 | 626 |
qed |
627 |
||
47694 | 628 |
lemma simple_integral_mono_AE: |
56949 | 629 |
assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g" |
47694 | 630 |
and mono: "AE x in M. f x \<le> g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
631 |
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g" |
40859 | 632 |
proof - |
56949 | 633 |
let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}" |
634 |
have "integral\<^sup>S M f = (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. fst y * ?\<mu> (\<lambda>x. (f x, g x) = y))" |
|
635 |
using f g by (intro simple_function_partition) auto |
|
636 |
also have "\<dots> \<le> (\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * ?\<mu> (\<lambda>x. (f x, g x) = y))" |
|
637 |
proof (clarsimp intro!: setsum_mono) |
|
40859 | 638 |
fix x assume "x \<in> space M" |
56949 | 639 |
let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)" |
640 |
show "f x * ?M \<le> g x * ?M" |
|
641 |
proof cases |
|
642 |
assume "?M \<noteq> 0" |
|
643 |
then have "0 < ?M" |
|
644 |
by (simp add: less_le emeasure_nonneg) |
|
645 |
also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)" |
|
646 |
using mono by (intro emeasure_mono_AE) auto |
|
647 |
finally have "\<not> \<not> f x \<le> g x" |
|
648 |
by (intro notI) auto |
|
649 |
then show ?thesis |
|
650 |
by (intro ereal_mult_right_mono) auto |
|
651 |
qed simp |
|
40859 | 652 |
qed |
56949 | 653 |
also have "\<dots> = integral\<^sup>S M g" |
654 |
using f g by (intro simple_function_partition[symmetric]) auto |
|
655 |
finally show ?thesis . |
|
40859 | 656 |
qed |
657 |
||
47694 | 658 |
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
|
659 |
assumes "simple_function M f" and "simple_function M g" |
38656 | 660 |
and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
661 |
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g" |
41705 | 662 |
using assms by (intro simple_integral_mono_AE) auto |
35582 | 663 |
|
47694 | 664 |
lemma simple_integral_cong_AE: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
665 |
assumes "simple_function M f" and "simple_function M g" |
47694 | 666 |
and "AE x in M. f x = g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
667 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
40859 | 668 |
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE) |
669 |
||
47694 | 670 |
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
|
671 |
assumes sf: "simple_function M f" "simple_function M g" |
47694 | 672 |
and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
673 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
40859 | 674 |
proof (intro simple_integral_cong_AE sf AE_I) |
47694 | 675 |
show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact |
40859 | 676 |
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M" |
677 |
using sf[THEN borel_measurable_simple_function] by auto |
|
678 |
qed simp |
|
679 |
||
47694 | 680 |
lemma simple_integral_indicator: |
56949 | 681 |
assumes A: "A \<in> sets M" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
682 |
assumes f: "simple_function M f" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
683 |
shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = |
56949 | 684 |
(\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))" |
685 |
proof - |
|
686 |
have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A" |
|
687 |
using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm) |
|
688 |
have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" |
|
689 |
by (auto simp: image_iff) |
|
690 |
||
691 |
have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = |
|
692 |
(\<Sum>y\<in>(\<lambda>x. (f x, indicator A x))`space M. (fst y * snd y) * emeasure M {x\<in>space M. (f x, indicator A x) = y})" |
|
693 |
using assms by (intro simple_function_partition) auto |
|
694 |
also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M. |
|
695 |
if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)" |
|
57418 | 696 |
by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum.cong) |
56949 | 697 |
also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, 1::ereal))`A. fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A))" |
57418 | 698 |
using assms by (subst setsum.If_cases) (auto intro!: simple_functionD(1) simp: eq) |
56949 | 699 |
also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))" |
57418 | 700 |
by (subst setsum.reindex [of fst]) (auto simp: inj_on_def) |
56949 | 701 |
also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))" |
702 |
using A[THEN sets.sets_into_space] |
|
57418 | 703 |
by (intro setsum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2) |
56949 | 704 |
finally show ?thesis . |
38656 | 705 |
qed |
35582 | 706 |
|
47694 | 707 |
lemma simple_integral_indicator_only[simp]: |
38656 | 708 |
assumes "A \<in> sets M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
709 |
shows "integral\<^sup>S M (indicator A) = emeasure M A" |
56949 | 710 |
using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms] |
711 |
by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm) |
|
35582 | 712 |
|
47694 | 713 |
lemma simple_integral_null_set: |
714 |
assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
715 |
shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0" |
38656 | 716 |
proof - |
47694 | 717 |
have "AE x in M. indicator N x = (0 :: ereal)" |
718 |
using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N]) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
719 |
then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
720 |
using assms apply (intro simple_integral_cong_AE) by auto |
40859 | 721 |
then show ?thesis by simp |
38656 | 722 |
qed |
35582 | 723 |
|
47694 | 724 |
lemma simple_integral_cong_AE_mult_indicator: |
725 |
assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
726 |
shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)" |
41705 | 727 |
using assms by (intro simple_integral_cong_AE) auto |
35582 | 728 |
|
47694 | 729 |
lemma simple_integral_cmult_indicator: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
730 |
assumes A: "A \<in> sets M" |
56949 | 731 |
shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * emeasure M A" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
732 |
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
|
733 |
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
|
734 |
|
56996 | 735 |
lemma simple_integral_nonneg: |
47694 | 736 |
assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
737 |
shows "0 \<le> integral\<^sup>S M f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
738 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
739 |
have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
740 |
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
|
741 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
742 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
743 |
|
56994 | 744 |
subsection {* Integral on nonnegative functions *} |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
745 |
|
56996 | 746 |
definition nn_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>N") where |
747 |
"integral\<^sup>N M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)" |
|
35692 | 748 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
749 |
syntax |
59357
f366643536cd
allow line breaks in integral notation
Andreas Lochbihler
parents:
59048
diff
changeset
|
750 |
"_nn_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+((2 _./ _)/ \<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
|
751 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
752 |
translations |
56996 | 753 |
"\<integral>\<^sup>+x. f \<partial>M" == "CONST nn_integral M (\<lambda>x. f)" |
40872 | 754 |
|
57025 | 755 |
lemma nn_integral_nonneg: "0 \<le> integral\<^sup>N M f" |
56996 | 756 |
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: nn_integral_def le_fun_def) |
40873 | 757 |
|
58606 | 758 |
lemma nn_integral_le_0[simp]: "integral\<^sup>N M f \<le> 0 \<longleftrightarrow> integral\<^sup>N M f = 0" |
759 |
using nn_integral_nonneg[of M f] by auto |
|
760 |
||
60064
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
761 |
lemma nn_integral_not_less_0 [simp]: "\<not> nn_integral M f < 0" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
762 |
by(simp add: not_less nn_integral_nonneg) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
763 |
|
56996 | 764 |
lemma nn_integral_not_MInfty[simp]: "integral\<^sup>N M f \<noteq> -\<infinity>" |
765 |
using nn_integral_nonneg[of M f] by auto |
|
47694 | 766 |
|
56996 | 767 |
lemma nn_integral_def_finite: |
768 |
"integral\<^sup>N M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^sup>S M g)" |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56213
diff
changeset
|
769 |
(is "_ = SUPREMUM ?A ?f") |
56996 | 770 |
unfolding nn_integral_def |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
771 |
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
|
772 |
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
|
773 |
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
|
774 |
note gM = g(1)[THEN borel_measurable_simple_function] |
50252 | 775 |
have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto |
46731 | 776 |
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
|
777 |
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
|
778 |
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
|
779 |
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
|
780 |
done |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56213
diff
changeset
|
781 |
show "integral\<^sup>S M g \<le> SUPREMUM ?A ?f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
782 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
783 |
have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto |
47694 | 784 |
assume "(emeasure M) ?G = 0" |
785 |
with gM have "AE x in M. x \<notin> ?G" |
|
786 |
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
|
787 |
with gM g show ?thesis |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
788 |
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
|
789 |
(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
|
790 |
next |
50252 | 791 |
assume \<mu>_G: "(emeasure M) ?G \<noteq> 0" |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56213
diff
changeset
|
792 |
have "SUPREMUM ?A (integral\<^sup>S M) = \<infinity>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
793 |
proof (intro SUP_PInfty) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
794 |
fix n :: nat |
47694 | 795 |
let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)" |
50252 | 796 |
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
|
797 |
then have "?g ?y \<in> ?A" by (rule g_in_A) |
47694 | 798 |
have "real n \<le> ?y * (emeasure M) ?G" |
50252 | 799 |
using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
800 |
also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
801 |
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
|
802 |
by (subst simple_integral_cmult_indicator) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
803 |
also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
804 |
by (intro simple_integral_mono) auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
805 |
finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
806 |
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
|
807 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
808 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
809 |
qed |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
810 |
qed (auto intro: SUP_upper) |
40873 | 811 |
|
56996 | 812 |
lemma nn_integral_mono_AE: |
813 |
assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v" |
|
814 |
unfolding nn_integral_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
815 |
proof (safe intro!: SUP_mono) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
816 |
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
|
817 |
from ae[THEN AE_E] guess N . note N = this |
47694 | 818 |
then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in) |
46731 | 819 |
let ?n = "\<lambda>x. n x * indicator (space M - N) x" |
47694 | 820 |
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
|
821 |
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
|
822 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
823 |
{ 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
|
824 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
825 |
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
|
826 |
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
|
827 |
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
|
828 |
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
|
829 |
qed simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
830 |
then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
831 |
moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
832 |
using ae_N N n by (auto intro!: simple_integral_mono_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
833 |
ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^sup>S M n \<le> integral\<^sup>S M m" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
834 |
by force |
38656 | 835 |
qed |
836 |
||
56996 | 837 |
lemma nn_integral_mono: |
838 |
"(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>N M u \<le> integral\<^sup>N M v" |
|
839 |
by (auto intro: nn_integral_mono_AE) |
|
40859 | 840 |
|
56996 | 841 |
lemma nn_integral_cong_AE: |
842 |
"AE x in M. u x = v x \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v" |
|
843 |
by (auto simp: eq_iff intro!: nn_integral_mono_AE) |
|
40859 | 844 |
|
56996 | 845 |
lemma nn_integral_cong: |
846 |
"(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v" |
|
847 |
by (auto intro: nn_integral_cong_AE) |
|
40859 | 848 |
|
59426
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
849 |
lemma nn_integral_cong_simp: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
850 |
"(\<And>x. x \<in> space M =simp=> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
851 |
by (auto intro: nn_integral_cong simp: simp_implies_def) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
852 |
|
56996 | 853 |
lemma nn_integral_cong_strong: |
854 |
"M = N \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N N v" |
|
855 |
by (auto intro: nn_integral_cong) |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
856 |
|
56996 | 857 |
lemma nn_integral_eq_simple_integral: |
858 |
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>N M f = integral\<^sup>S M f" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
859 |
proof - |
46731 | 860 |
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
|
861 |
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
|
862 |
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
|
863 |
by (auto simp: fun_eq_iff max_def split: split_indicator) |
56996 | 864 |
have "integral\<^sup>N M ?f \<le> integral\<^sup>S M ?f" using f' |
865 |
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def nn_integral_def) |
|
866 |
moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>N M ?f" |
|
867 |
unfolding nn_integral_def |
|
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
868 |
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
|
869 |
ultimately show ?thesis |
56996 | 870 |
by (simp cong: nn_integral_cong simple_integral_cong) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
871 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
872 |
|
56996 | 873 |
lemma nn_integral_eq_simple_integral_AE: |
874 |
assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>N M f = integral\<^sup>S M f" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
875 |
proof - |
47694 | 876 |
have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max) |
56996 | 877 |
with f have "integral\<^sup>N M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))" |
878 |
by (simp cong: nn_integral_cong_AE simple_integral_cong_AE |
|
879 |
add: nn_integral_eq_simple_integral) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
880 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
881 |
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
|
882 |
qed |
40873 | 883 |
|
56996 | 884 |
lemma nn_integral_SUP_approx: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
885 |
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
|
886 |
and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}" |
56996 | 887 |
shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>N M (f i))" (is "_ \<le> ?S") |
43920 | 888 |
proof (rule ereal_le_mult_one_interval) |
56996 | 889 |
have "0 \<le> (SUP i. integral\<^sup>N M (f i))" |
890 |
using f(3) by (auto intro!: SUP_upper2 nn_integral_nonneg) |
|
891 |
then show "(SUP i. integral\<^sup>N M (f i)) \<noteq> -\<infinity>" by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
892 |
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
|
893 |
using u(3) by auto |
43920 | 894 |
fix a :: ereal assume "0 < a" "a < 1" |
38656 | 895 |
hence "a \<noteq> 0" by auto |
46731 | 896 |
let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}" |
38656 | 897 |
have B: "\<And>i. ?B i \<in> sets M" |
56949 | 898 |
using f `simple_function M u`[THEN borel_measurable_simple_function] by auto |
38656 | 899 |
|
46731 | 900 |
let ?uB = "\<lambda>i x. u x * indicator (?B i) x" |
38656 | 901 |
|
902 |
{ fix i have "?B i \<subseteq> ?B (Suc i)" |
|
903 |
proof safe |
|
904 |
fix i x assume "a * u x \<le> f i x" |
|
905 |
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
|
906 |
using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto |
38656 | 907 |
finally show "a * u x \<le> f (Suc i) x" . |
908 |
qed } |
|
909 |
note B_mono = this |
|
35582 | 910 |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
911 |
note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B] |
38656 | 912 |
|
46731 | 913 |
let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n" |
47694 | 914 |
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
|
915 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
916 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
917 |
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
|
918 |
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
|
919 |
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
|
920 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
921 |
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
|
922 |
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
|
923 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
924 |
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
|
925 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
926 |
assume "u x \<noteq> 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
927 |
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
|
928 |
have "a * u x < 1 * u x" |
43920 | 929 |
by (intro ereal_mult_strict_right_mono) (auto simp: image_iff) |
46884 | 930 |
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
|
931 |
finally obtain i where "a * u x < f i x" unfolding SUP_def |
56166 | 932 |
by (auto simp add: less_SUP_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
933 |
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
|
934 |
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
|
935 |
qed |
40859 | 936 |
qed |
47694 | 937 |
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
|
938 |
qed |
38656 | 939 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
940 |
have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))" |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
941 |
unfolding simple_integral_indicator[OF B `simple_function M u`] |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
942 |
proof (subst SUP_ereal_setsum, safe) |
38656 | 943 |
fix x n assume "x \<in> space M" |
47694 | 944 |
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)" |
945 |
using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff) |
|
38656 | 946 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
947 |
show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
948 |
using measure_conv u_range B_u unfolding simple_integral_def |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
949 |
by (auto intro!: setsum.cong SUP_ereal_mult_left [symmetric]) |
38656 | 950 |
qed |
951 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
952 |
have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
953 |
apply (subst SUP_ereal_mult_left [symmetric]) |
38705 | 954 |
proof (safe intro!: SUP_mono bexI) |
38656 | 955 |
fix i |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
956 |
have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
957 |
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
|
958 |
by (subst simple_integral_mult) (auto split: split_indicator) |
56996 | 959 |
also have "\<dots> \<le> integral\<^sup>N M (f i)" |
38656 | 960 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
961 |
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
|
962 |
show ?thesis using f(3) * u_range `0 < a` |
56996 | 963 |
by (subst nn_integral_eq_simple_integral[symmetric]) |
964 |
(auto intro!: nn_integral_mono split: split_indicator) |
|
38656 | 965 |
qed |
56996 | 966 |
finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>N M (f i)" |
38656 | 967 |
by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
968 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
969 |
fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range |
56996 | 970 |
by (intro simple_integral_nonneg) (auto split: split_indicator) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
971 |
qed (insert `0 < a`, auto) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
972 |
ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp |
35582 | 973 |
qed |
974 |
||
56996 | 975 |
lemma incseq_nn_integral: |
976 |
assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>N M (f i))" |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
977 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
978 |
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
|
979 |
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
|
980 |
then show ?thesis |
56996 | 981 |
by (auto intro!: incseq_SucI nn_integral_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
982 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
983 |
|
59000 | 984 |
lemma nn_integral_max_0: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>N M f" |
985 |
by (simp add: le_fun_def nn_integral_def) |
|
986 |
||
35582 | 987 |
text {* Beppo-Levi monotone convergence theorem *} |
56996 | 988 |
lemma nn_integral_monotone_convergence_SUP: |
59000 | 989 |
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" |
56996 | 990 |
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
991 |
proof (rule antisym) |
56996 | 992 |
show "(SUP j. integral\<^sup>N M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)" |
993 |
by (auto intro!: SUP_least SUP_upper nn_integral_mono) |
|
38656 | 994 |
next |
59000 | 995 |
have f': "incseq (\<lambda>i x. max 0 (f i x))" |
996 |
using f by (auto simp: incseq_def le_fun_def not_le split: split_max) |
|
997 |
(blast intro: order_trans less_imp_le) |
|
998 |
have "(\<integral>\<^sup>+ x. max 0 (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. max 0 (f i x)) \<partial>M)" |
|
999 |
unfolding sup_max[symmetric] Complete_Lattices.SUP_sup_distrib[symmetric] by simp |
|
1000 |
also have "\<dots> \<le> (SUP i. (\<integral>\<^sup>+ x. max 0 (f i x) \<partial>M))" |
|
1001 |
unfolding nn_integral_def_finite[of _ "\<lambda>x. SUP i. max 0 (f i x)"] |
|
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
1002 |
proof (safe intro!: SUP_least) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1003 |
fix g assume g: "simple_function M g" |
59000 | 1004 |
and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. max 0 (f i x))" "range g \<subseteq> {0..<\<infinity>}" |
1005 |
then have "\<And>x. 0 \<le> (SUP i. max 0 (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
|
1006 |
using f by (auto intro!: SUP_upper2) |
59000 | 1007 |
with * show "integral\<^sup>S M g \<le> (SUP j. \<integral>\<^sup>+x. max 0 (f j x) \<partial>M)" |
1008 |
by (intro nn_integral_SUP_approx[OF f' _ _ g _ g']) |
|
1009 |
(auto simp: le_fun_def max_def intro!: measurable_If f borel_measurable_le) |
|
35582 | 1010 |
qed |
59000 | 1011 |
finally show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>N M (f j))" |
1012 |
unfolding nn_integral_max_0 . |
|
35582 | 1013 |
qed |
1014 |
||
56996 | 1015 |
lemma nn_integral_monotone_convergence_SUP_AE: |
47694 | 1016 |
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" |
56996 | 1017 |
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))" |
40859 | 1018 |
proof - |
47694 | 1019 |
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
|
1020 |
by (simp add: AE_all_countable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1021 |
from this[THEN AE_E] guess N . note N = this |
46731 | 1022 |
let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0" |
47694 | 1023 |
have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N]) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1024 |
then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)" |
56996 | 1025 |
by (auto intro!: nn_integral_cong_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1026 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))" |
56996 | 1027 |
proof (rule nn_integral_monotone_convergence_SUP) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1028 |
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
|
1029 |
{ fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M" |
59000 | 1030 |
using f N(3) by (intro measurable_If_set) auto } |
40859 | 1031 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1032 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))" |
56996 | 1033 |
using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] nn_integral_cong_AE ext) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1034 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1035 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1036 |
|
56996 | 1037 |
lemma nn_integral_monotone_convergence_SUP_AE_incseq: |
47694 | 1038 |
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" |
56996 | 1039 |
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1040 |
using f[unfolded incseq_Suc_iff le_fun_def] |
56996 | 1041 |
by (intro nn_integral_monotone_convergence_SUP_AE[OF _ borel]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1042 |
auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1043 |
|
56996 | 1044 |
lemma nn_integral_monotone_convergence_simple: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1045 |
assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1046 |
shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)" |
56996 | 1047 |
using assms unfolding nn_integral_monotone_convergence_SUP[OF f(1) |
59000 | 1048 |
f(3)[THEN borel_measurable_simple_function]] |
56996 | 1049 |
by (auto intro!: nn_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPREMUM UNIV"] ext) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1050 |
|
56996 | 1051 |
lemma nn_integral_cong_pos: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1052 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x" |
56996 | 1053 |
shows "integral\<^sup>N M f = integral\<^sup>N M g" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1054 |
proof - |
56996 | 1055 |
have "integral\<^sup>N M (\<lambda>x. max 0 (f x)) = integral\<^sup>N M (\<lambda>x. max 0 (g x))" |
1056 |
proof (intro nn_integral_cong) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1057 |
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
|
1058 |
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
|
1059 |
by (auto split: split_max) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1060 |
qed |
56996 | 1061 |
then show ?thesis by (simp add: nn_integral_max_0) |
40859 | 1062 |
qed |
1063 |
||
47694 | 1064 |
lemma SUP_simple_integral_sequences: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1065 |
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
|
1066 |
and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)" |
47694 | 1067 |
and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1068 |
shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))" |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56213
diff
changeset
|
1069 |
(is "SUPREMUM _ ?F = SUPREMUM _ ?G") |
38656 | 1070 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1071 |
have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)" |
56996 | 1072 |
using f by (rule nn_integral_monotone_convergence_simple) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1073 |
also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)" |
56996 | 1074 |
unfolding eq[THEN nn_integral_cong_AE] .. |
38656 | 1075 |
also have "\<dots> = (SUP i. ?G i)" |
56996 | 1076 |
using g by (rule nn_integral_monotone_convergence_simple[symmetric]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1077 |
finally show ?thesis by simp |
38656 | 1078 |
qed |
1079 |
||
56996 | 1080 |
lemma nn_integral_const[simp]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1081 |
"0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)" |
56996 | 1082 |
by (subst nn_integral_eq_simple_integral) auto |
38656 | 1083 |
|
56996 | 1084 |
lemma nn_integral_linear: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1085 |
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
|
1086 |
and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x" |
56996 | 1087 |
shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>N M f + integral\<^sup>N M g" |
1088 |
(is "integral\<^sup>N M ?L = _") |
|
35582 | 1089 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1090 |
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u . |
56996 | 1091 |
note u = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1092 |
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v . |
56996 | 1093 |
note v = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this |
46731 | 1094 |
let ?L' = "\<lambda>i x. a * u i x + v i x" |
38656 | 1095 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1096 |
have "?L \<in> borel_measurable M" using assms by auto |
38656 | 1097 |
from borel_measurable_implies_simple_function_sequence'[OF this] guess l . |
56996 | 1098 |
note l = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1099 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1100 |
have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1101 |
using u v `0 \<le> a` |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1102 |
by (auto simp: incseq_Suc_iff le_fun_def |
43920 | 1103 |
intro!: add_mono ereal_mult_left_mono simple_integral_mono) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1104 |
have pos: "\<And>i. 0 \<le> integral\<^sup>S M (u i)" "\<And>i. 0 \<le> integral\<^sup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^sup>S M (u i)" |
56996 | 1105 |
using u v `0 \<le> a` by (auto simp: simple_integral_nonneg) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1106 |
{ fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1107 |
by (auto split: split_if_asm) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1108 |
note not_MInf = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1109 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1110 |
have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1111 |
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
|
1112 |
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
|
1113 |
using u v `0 \<le> a` unfolding incseq_Suc_iff le_fun_def |
56537 | 1114 |
by (auto intro!: add_mono ereal_mult_left_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1115 |
{ fix x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1116 |
{ 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
|
1117 |
by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1118 |
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
|
1119 |
using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
1120 |
by (subst SUP_ereal_mult_left [symmetric, OF _ u(6) `0 \<le> a`]) |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1121 |
(auto intro!: SUP_ereal_add |
56537 | 1122 |
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) } |
47694 | 1123 |
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
|
1124 |
unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2) |
56537 | 1125 |
by (intro AE_I2) (auto split: split_max) |
38656 | 1126 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1127 |
also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))" |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56213
diff
changeset
|
1128 |
using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1129 |
finally have "(\<integral>\<^sup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^sup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+x. max 0 (g x) \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1130 |
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
|
1131 |
unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
1132 |
apply (subst SUP_ereal_mult_left [symmetric, OF _ pos(1) `0 \<le> a`]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
1133 |
apply simp |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
1134 |
apply (subst SUP_ereal_add [symmetric, OF inc not_MInf]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
1135 |
. |
56996 | 1136 |
then show ?thesis by (simp add: nn_integral_max_0) |
38656 | 1137 |
qed |
1138 |
||
56996 | 1139 |
lemma nn_integral_cmult: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1140 |
assumes f: "f \<in> borel_measurable M" "0 \<le> c" |
56996 | 1141 |
shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>N M f" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1142 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1143 |
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c` |
43920 | 1144 |
by (auto split: split_max simp: ereal_zero_le_0_iff) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1145 |
have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)" |
56996 | 1146 |
by (simp add: nn_integral_max_0) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1147 |
then show ?thesis |
56996 | 1148 |
using nn_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f |
1149 |
by (auto simp: nn_integral_max_0) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1150 |
qed |
38656 | 1151 |
|
56996 | 1152 |
lemma nn_integral_multc: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1153 |
assumes "f \<in> borel_measurable M" "0 \<le> c" |
56996 | 1154 |
shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>N M f * c" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
1155 |
unfolding mult.commute[of _ c] nn_integral_cmult[OF assms] by simp |
41096 | 1156 |
|
59000 | 1157 |
lemma nn_integral_divide: |
1158 |
"0 < c \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> (\<integral>\<^sup>+x. f x / c \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M) / c" |
|
1159 |
unfolding divide_ereal_def |
|
1160 |
apply (rule nn_integral_multc) |
|
1161 |
apply assumption |
|
1162 |
apply (cases c) |
|
1163 |
apply auto |
|
1164 |
done |
|
1165 |
||
56996 | 1166 |
lemma nn_integral_indicator[simp]: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1167 |
"A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A" |
56996 | 1168 |
by (subst nn_integral_eq_simple_integral) |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1169 |
(auto simp: simple_integral_indicator) |
38656 | 1170 |
|
56996 | 1171 |
lemma nn_integral_cmult_indicator: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1172 |
"0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A" |
56996 | 1173 |
by (subst nn_integral_eq_simple_integral) |
41544 | 1174 |
(auto simp: simple_function_indicator simple_integral_indicator) |
38656 | 1175 |
|
56996 | 1176 |
lemma nn_integral_indicator': |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1177 |
assumes [measurable]: "A \<inter> space M \<in> sets M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1178 |
shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)" |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1179 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1180 |
have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)" |
56996 | 1181 |
by (intro nn_integral_cong) (simp split: split_indicator) |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1182 |
also have "\<dots> = emeasure M (A \<inter> space M)" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1183 |
by simp |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1184 |
finally show ?thesis . |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1185 |
qed |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1186 |
|
56996 | 1187 |
lemma nn_integral_add: |
47694 | 1188 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
1189 |
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
56996 | 1190 |
shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>N M f + integral\<^sup>N M g" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1191 |
proof - |
47694 | 1192 |
have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)" |
56537 | 1193 |
using assms by (auto split: split_max) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1194 |
have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)" |
56996 | 1195 |
by (simp add: nn_integral_max_0) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1196 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)" |
56996 | 1197 |
unfolding ae[THEN nn_integral_cong_AE] .. |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1198 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)" |
56996 | 1199 |
using nn_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
|
1200 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1201 |
finally show ?thesis |
56996 | 1202 |
by (simp add: nn_integral_max_0) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1203 |
qed |
38656 | 1204 |
|
56996 | 1205 |
lemma nn_integral_setsum: |
47694 | 1206 |
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" |
56996 | 1207 |
shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>N M (f i))" |
38656 | 1208 |
proof cases |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1209 |
assume f: "finite P" |
47694 | 1210 |
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
|
1211 |
from f this assms(1) show ?thesis |
38656 | 1212 |
proof induct |
1213 |
case (insert i P) |
|
47694 | 1214 |
then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x" |
1215 |
"(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1216 |
by (auto intro!: setsum_nonneg) |
56996 | 1217 |
from nn_integral_add[OF this] |
38656 | 1218 |
show ?case using insert by auto |
1219 |
qed simp |
|
1220 |
qed simp |
|
1221 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1222 |
lemma nn_integral_bound_simple_function: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1223 |
assumes bnd: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> space M \<Longrightarrow> f x < \<infinity>" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1224 |
assumes f[measurable]: "simple_function M f" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1225 |
assumes supp: "emeasure M {x\<in>space M. f x \<noteq> 0} < \<infinity>" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1226 |
shows "nn_integral M f < \<infinity>" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1227 |
proof cases |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1228 |
assume "space M = {}" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1229 |
then have "nn_integral M f = (\<integral>\<^sup>+x. 0 \<partial>M)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1230 |
by (intro nn_integral_cong) auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1231 |
then show ?thesis by simp |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1232 |
next |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1233 |
assume "space M \<noteq> {}" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1234 |
with simple_functionD(1)[OF f] bnd have bnd: "0 \<le> Max (f`space M) \<and> Max (f`space M) < \<infinity>" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1235 |
by (subst Max_less_iff) (auto simp: Max_ge_iff) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1236 |
|
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1237 |
have "nn_integral M f \<le> (\<integral>\<^sup>+x. Max (f`space M) * indicator {x\<in>space M. f x \<noteq> 0} x \<partial>M)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1238 |
proof (rule nn_integral_mono) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1239 |
fix x assume "x \<in> space M" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1240 |
with f show "f x \<le> Max (f ` space M) * indicator {x \<in> space M. f x \<noteq> 0} x" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1241 |
by (auto split: split_indicator intro!: Max_ge simple_functionD) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1242 |
qed |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1243 |
also have "\<dots> < \<infinity>" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1244 |
using bnd supp by (subst nn_integral_cmult) auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1245 |
finally show ?thesis . |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1246 |
qed |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57418
diff
changeset
|
1247 |
|
56996 | 1248 |
lemma nn_integral_Markov_inequality: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1249 |
assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1250 |
shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)" |
47694 | 1251 |
(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
|
1252 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1253 |
have "?A \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1254 |
using `A \<in> sets M` u by auto |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1255 |
hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)" |
56996 | 1256 |
using nn_integral_indicator by simp |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1257 |
also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c |
56996 | 1258 |
by (auto intro!: nn_integral_mono_AE |
43920 | 1259 |
simp: indicator_def ereal_zero_le_0_iff) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1260 |
also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1261 |
using assms |
56996 | 1262 |
by (auto intro!: nn_integral_cmult simp: ereal_zero_le_0_iff) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1263 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1264 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1265 |
|
56996 | 1266 |
lemma nn_integral_noteq_infinite: |
47694 | 1267 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
56996 | 1268 |
and "integral\<^sup>N M g \<noteq> \<infinity>" |
47694 | 1269 |
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
|
1270 |
proof (rule ccontr) |
47694 | 1271 |
assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)" |
1272 |
have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0" |
|
1273 |
using c g by (auto simp add: AE_iff_null) |
|
1274 |
moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets) |
|
1275 |
ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto |
|
1276 |
then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1277 |
also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)" |
56996 | 1278 |
using g by (subst nn_integral_cmult_indicator) auto |
1279 |
also have "\<dots> \<le> integral\<^sup>N M g" |
|
1280 |
using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def) |
|
1281 |
finally show False using `integral\<^sup>N M g \<noteq> \<infinity>` by auto |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1282 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1283 |
|
56996 | 1284 |
lemma nn_integral_PInf: |
56949 | 1285 |
assumes f: "f \<in> borel_measurable M" |
56996 | 1286 |
and not_Inf: "integral\<^sup>N M f \<noteq> \<infinity>" |
56949 | 1287 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
1288 |
proof - |
|
1289 |
have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)" |
|
56996 | 1290 |
using f by (subst nn_integral_cmult_indicator) (auto simp: measurable_sets) |
1291 |
also have "\<dots> \<le> integral\<^sup>N M (\<lambda>x. max 0 (f x))" |
|
1292 |
by (auto intro!: nn_integral_mono simp: indicator_def max_def) |
|
1293 |
finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>N M f" |
|
1294 |
by (simp add: nn_integral_max_0) |
|
56949 | 1295 |
moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)" |
1296 |
by (rule emeasure_nonneg) |
|
1297 |
ultimately show ?thesis |
|
1298 |
using assms by (auto split: split_if_asm) |
|
1299 |
qed |
|
1300 |
||
56996 | 1301 |
lemma nn_integral_PInf_AE: |
1302 |
assumes "f \<in> borel_measurable M" "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>" |
|
56949 | 1303 |
proof (rule AE_I) |
1304 |
show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
|
56996 | 1305 |
by (rule nn_integral_PInf[OF assms]) |
56949 | 1306 |
show "f -` {\<infinity>} \<inter> space M \<in> sets M" |
1307 |
using assms by (auto intro: borel_measurable_vimage) |
|
1308 |
qed auto |
|
1309 |
||
1310 |
lemma simple_integral_PInf: |
|
1311 |
assumes "simple_function M f" "\<And>x. 0 \<le> f x" |
|
1312 |
and "integral\<^sup>S M f \<noteq> \<infinity>" |
|
1313 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
|
56996 | 1314 |
proof (rule nn_integral_PInf) |
56949 | 1315 |
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function) |
56996 | 1316 |
show "integral\<^sup>N M f \<noteq> \<infinity>" |
1317 |
using assms by (simp add: nn_integral_eq_simple_integral) |
|
56949 | 1318 |
qed |
1319 |
||
56996 | 1320 |
lemma nn_integral_diff: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1321 |
assumes f: "f \<in> borel_measurable M" |
47694 | 1322 |
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
56996 | 1323 |
and fin: "integral\<^sup>N M g \<noteq> \<infinity>" |
47694 | 1324 |
and mono: "AE x in M. g x \<le> f x" |
56996 | 1325 |
shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>N M f - integral\<^sup>N M g" |
38656 | 1326 |
proof - |
47694 | 1327 |
have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x" |
43920 | 1328 |
using assms by (auto intro: ereal_diff_positive) |
47694 | 1329 |
have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto |
43920 | 1330 |
{ fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b" |
1331 |
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
|
1332 |
note * = this |
47694 | 1333 |
then have "AE x in M. f x = f x - g x + g x" |
56996 | 1334 |
using mono nn_integral_noteq_infinite[OF g fin] assms by auto |
1335 |
then have **: "integral\<^sup>N M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>N M g" |
|
1336 |
unfolding nn_integral_add[OF diff g, symmetric] |
|
1337 |
by (rule nn_integral_cong_AE) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1338 |
show ?thesis unfolding ** |
56996 | 1339 |
using fin nn_integral_nonneg[of M g] |
1340 |
by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>N M g"]) auto |
|
38656 | 1341 |
qed |
1342 |
||
56996 | 1343 |
lemma nn_integral_suminf: |
47694 | 1344 |
assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x" |
56996 | 1345 |
shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>N M (f i))" |
38656 | 1346 |
proof - |
47694 | 1347 |
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
|
1348 |
using assms by (auto simp: AE_all_countable) |
56996 | 1349 |
have "(\<Sum>i. integral\<^sup>N M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>N M (f i))" |
1350 |
using nn_integral_nonneg by (rule suminf_ereal_eq_SUP) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1351 |
also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)" |
56996 | 1352 |
unfolding nn_integral_setsum[OF f] .. |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1353 |
also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos |
56996 | 1354 |
by (intro nn_integral_monotone_convergence_SUP_AE[symmetric]) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1355 |
(elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1356 |
also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos |
56996 | 1357 |
by (intro nn_integral_cong_AE) (auto simp: suminf_ereal_eq_SUP) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1358 |
finally show ?thesis by simp |
38656 | 1359 |
qed |
1360 |
||
56996 | 1361 |
lemma nn_integral_mult_bounded_inf: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1362 |
assumes f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1363 |
and c: "0 \<le> c" "c \<noteq> \<infinity>" and ae: "AE x in M. g x \<le> c * f x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1364 |
shows "(\<integral>\<^sup>+x. g x \<partial>M) < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1365 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1366 |
have "(\<integral>\<^sup>+x. g x \<partial>M) \<le> (\<integral>\<^sup>+x. c * f x \<partial>M)" |
56996 | 1367 |
by (intro nn_integral_mono_AE ae) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1368 |
also have "(\<integral>\<^sup>+x. c * f x \<partial>M) < \<infinity>" |
56996 | 1369 |
using c f by (subst nn_integral_cmult) auto |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1370 |
finally show ?thesis . |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1371 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1372 |
|
38656 | 1373 |
text {* Fatou's lemma: convergence theorem on limes inferior *} |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1374 |
|
56996 | 1375 |
lemma nn_integral_liminf: |
43920 | 1376 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
47694 | 1377 |
assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x" |
56996 | 1378 |
shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))" |
38656 | 1379 |
proof - |
47694 | 1380 |
have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1381 |
have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1382 |
(SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)" |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1383 |
unfolding liminf_SUP_INF using pos u |
56996 | 1384 |
by (intro nn_integral_monotone_convergence_SUP_AE) |
44937
22c0857b8aab
removed further legacy rules from Complete_Lattices
hoelzl
parents:
44928
diff
changeset
|
1385 |
(elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono) |
56996 | 1386 |
also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))" |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1387 |
unfolding liminf_SUP_INF |
56996 | 1388 |
by (auto intro!: SUP_mono exI INF_greatest nn_integral_mono INF_lower) |
38656 | 1389 |
finally show ?thesis . |
35582 | 1390 |
qed |
1391 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1392 |
lemma le_Limsup: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1393 |
"F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. c \<le> g x) F \<Longrightarrow> c \<le> Limsup F g" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1394 |
using Limsup_mono[of "\<lambda>_. c" g F] by (simp add: Limsup_const) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1395 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1396 |
lemma Limsup_le: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1397 |
"F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. f x \<le> c) F \<Longrightarrow> Limsup F f \<le> c" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1398 |
using Limsup_mono[of f "\<lambda>_. c" F] by (simp add: Limsup_const) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1399 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1400 |
lemma ereal_mono_minus_cancel: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1401 |
fixes a b c :: ereal |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1402 |
shows "c - a \<le> c - b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> b \<le> a" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1403 |
by (cases a b c rule: ereal3_cases) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1404 |
|
56996 | 1405 |
lemma nn_integral_limsup: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1406 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1407 |
assumes [measurable]: "\<And>i. u i \<in> borel_measurable M" "w \<in> borel_measurable M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1408 |
assumes bounds: "\<And>i. AE x in M. 0 \<le> u i x" "\<And>i. AE x in M. u i x \<le> w x" and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>" |
56996 | 1409 |
shows "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)" |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1410 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1411 |
have bnd: "AE x in M. \<forall>i. 0 \<le> u i x \<and> u i x \<le> w x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1412 |
using bounds by (auto simp: AE_all_countable) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1413 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1414 |
from bounds[of 0] have w_nonneg: "AE x in M. 0 \<le> w x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1415 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1416 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1417 |
have "(\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+x. w x - limsup (\<lambda>n. u n x) \<partial>M)" |
56996 | 1418 |
proof (intro nn_integral_diff[symmetric]) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1419 |
show "AE x in M. 0 \<le> limsup (\<lambda>n. u n x)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1420 |
using bnd by (auto intro!: le_Limsup) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1421 |
show "AE x in M. limsup (\<lambda>n. u n x) \<le> w x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1422 |
using bnd by (auto intro!: Limsup_le) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1423 |
then have "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) < \<infinity>" |
56996 | 1424 |
by (intro nn_integral_mult_bounded_inf[OF _ w, of 1]) auto |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1425 |
then show "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1426 |
by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1427 |
qed auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1428 |
also have "\<dots> = (\<integral>\<^sup>+x. liminf (\<lambda>n. w x - u n x) \<partial>M)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1429 |
using w_nonneg |
56996 | 1430 |
by (intro nn_integral_cong_AE, eventually_elim) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1431 |
(auto intro!: liminf_ereal_cminus[symmetric]) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1432 |
also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M)" |
56996 | 1433 |
proof (rule nn_integral_liminf) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1434 |
fix i show "AE x in M. 0 \<le> w x - u i x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1435 |
using bounds[of i] by eventually_elim (auto intro: ereal_diff_positive) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1436 |
qed simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1437 |
also have "(\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M) = (\<lambda>n. (\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. u n x \<partial>M))" |
56996 | 1438 |
proof (intro ext nn_integral_diff) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1439 |
fix i have "(\<integral>\<^sup>+x. u i x \<partial>M) < \<infinity>" |
56996 | 1440 |
using bounds by (intro nn_integral_mult_bounded_inf[OF _ w, of 1]) auto |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1441 |
then show "(\<integral>\<^sup>+x. u i x \<partial>M) \<noteq> \<infinity>" by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1442 |
qed (insert bounds, auto) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1443 |
also have "liminf (\<lambda>n. (\<integral>\<^sup>+x. w x \<partial>M) - (\<integral>\<^sup>+x. u n x \<partial>M)) = (\<integral>\<^sup>+x. w x \<partial>M) - limsup (\<lambda>n. \<integral>\<^sup>+x. u n x \<partial>M)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1444 |
using w by (intro liminf_ereal_cminus) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1445 |
finally show ?thesis |
56996 | 1446 |
by (rule ereal_mono_minus_cancel) (intro w nn_integral_nonneg)+ |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1447 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1448 |
|
57025 | 1449 |
lemma nn_integral_LIMSEQ: |
1450 |
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>n x. 0 \<le> f n x" |
|
1451 |
and u: "\<And>x. (\<lambda>i. f i x) ----> u x" |
|
1452 |
shows "(\<lambda>n. integral\<^sup>N M (f n)) ----> integral\<^sup>N M u" |
|
1453 |
proof - |
|
1454 |
have "(\<lambda>n. integral\<^sup>N M (f n)) ----> (SUP n. integral\<^sup>N M (f n))" |
|
1455 |
using f by (intro LIMSEQ_SUP[of "\<lambda>n. integral\<^sup>N M (f n)"] incseq_nn_integral) |
|
1456 |
also have "(SUP n. integral\<^sup>N M (f n)) = integral\<^sup>N M (\<lambda>x. SUP n. f n x)" |
|
1457 |
using f by (intro nn_integral_monotone_convergence_SUP[symmetric]) |
|
1458 |
also have "integral\<^sup>N M (\<lambda>x. SUP n. f n x) = integral\<^sup>N M (\<lambda>x. u x)" |
|
1459 |
using f by (subst SUP_Lim_ereal[OF _ u]) (auto simp: incseq_def le_fun_def) |
|
1460 |
finally show ?thesis . |
|
1461 |
qed |
|
1462 |
||
56996 | 1463 |
lemma nn_integral_dominated_convergence: |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1464 |
assumes [measurable]: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1465 |
"\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1466 |
and bound: "\<And>j. AE x in M. 0 \<le> u j x" "\<And>j. AE x in M. u j x \<le> w x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1467 |
and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1468 |
and u': "AE x in M. (\<lambda>i. u i x) ----> u' x" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1469 |
shows "(\<lambda>i. (\<integral>\<^sup>+x. u i x \<partial>M)) ----> (\<integral>\<^sup>+x. u' x \<partial>M)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1470 |
proof - |
56996 | 1471 |
have "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)" |
1472 |
by (intro nn_integral_limsup[OF _ _ bound w]) auto |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1473 |
moreover have "(\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)" |
56996 | 1474 |
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot) |
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1475 |
moreover have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)" |
56996 | 1476 |
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot) |
1477 |
moreover have "(\<integral>\<^sup>+x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))" |
|
1478 |
by (intro nn_integral_liminf[OF _ bound(1)]) auto |
|
1479 |
moreover have "liminf (\<lambda>n. integral\<^sup>N M (u n)) \<le> limsup (\<lambda>n. integral\<^sup>N M (u n))" |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1480 |
by (intro Liminf_le_Limsup sequentially_bot) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1481 |
ultimately show ?thesis |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1482 |
by (intro Liminf_eq_Limsup) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1483 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1484 |
|
56996 | 1485 |
lemma nn_integral_null_set: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1486 |
assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0" |
38656 | 1487 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1488 |
have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
56996 | 1489 |
proof (intro nn_integral_cong_AE AE_I) |
40859 | 1490 |
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N" |
1491 |
by (auto simp: indicator_def) |
|
47694 | 1492 |
show "(emeasure M) N = 0" "N \<in> sets M" |
40859 | 1493 |
using assms by auto |
35582 | 1494 |
qed |
40859 | 1495 |
then show ?thesis by simp |
38656 | 1496 |
qed |
35582 | 1497 |
|
56996 | 1498 |
lemma nn_integral_0_iff: |
47694 | 1499 |
assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x" |
56996 | 1500 |
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0" |
47694 | 1501 |
(is "_ \<longleftrightarrow> (emeasure M) ?A = 0") |
35582 | 1502 |
proof - |
56996 | 1503 |
have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>N M u" |
1504 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
|
38656 | 1505 |
show ?thesis |
1506 |
proof |
|
47694 | 1507 |
assume "(emeasure M) ?A = 0" |
56996 | 1508 |
with nn_integral_null_set[of ?A M u] u |
1509 |
show "integral\<^sup>N M u = 0" by (simp add: u_eq null_sets_def) |
|
38656 | 1510 |
next |
43920 | 1511 |
{ fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r" |
1512 |
then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def) |
|
1513 |
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
|
1514 |
note gt_1 = this |
56996 | 1515 |
assume *: "integral\<^sup>N M u = 0" |
46731 | 1516 |
let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}" |
47694 | 1517 |
have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))" |
38656 | 1518 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1519 |
{ fix n :: nat |
56996 | 1520 |
from nn_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"] |
47694 | 1521 |
have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp |
1522 |
moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto |
|
1523 |
ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto } |
|
38656 | 1524 |
thus ?thesis by simp |
35582 | 1525 |
qed |
47694 | 1526 |
also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)" |
1527 |
proof (safe intro!: SUP_emeasure_incseq) |
|
38656 | 1528 |
fix n show "?M n \<inter> ?A \<in> sets M" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
1529 |
using u by (auto intro!: sets.Int) |
38656 | 1530 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1531 |
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
|
1532 |
proof (safe intro!: incseq_SucI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1533 |
fix n :: nat and x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1534 |
assume *: "1 \<le> real n * u x" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1535 |
also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x" |
43920 | 1536 |
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
|
1537 |
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
|
1538 |
qed |
38656 | 1539 |
qed |
47694 | 1540 |
also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}" |
1541 |
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
|
1542 |
fix x assume "0 < u x" and [simp, intro]: "x \<in> space M" |
38656 | 1543 |
show "x \<in> (\<Union>n. ?M n \<inter> ?A)" |
1544 |
proof (cases "u x") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1545 |
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
|
1546 |
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
|
1547 |
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
|
1548 |
hence "1 \<le> real j * r" using real `0 < r` by auto |
43920 | 1549 |
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
|
1550 |
qed (insert `0 < u x`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1551 |
qed auto |
47694 | 1552 |
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
|
1553 |
moreover |
47694 | 1554 |
from pos have "AE x in M. \<not> (u x < 0)" by auto |
1555 |
then have "(emeasure M) {x\<in>space M. u x < 0} = 0" |
|
1556 |
using AE_iff_null[of M] u by auto |
|
1557 |
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}" |
|
1558 |
using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"]) |
|
1559 |
ultimately show "(emeasure M) ?A = 0" by simp |
|
35582 | 1560 |
qed |
1561 |
qed |
|
1562 |
||
56996 | 1563 |
lemma nn_integral_0_iff_AE: |
41705 | 1564 |
assumes u: "u \<in> borel_measurable M" |
56996 | 1565 |
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)" |
41705 | 1566 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1567 |
have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M" |
41705 | 1568 |
using u by auto |
56996 | 1569 |
from nn_integral_0_iff[of "\<lambda>x. max 0 (u x)"] |
1570 |
have "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)" |
|
1571 |
unfolding nn_integral_max_0 |
|
47694 | 1572 |
using AE_iff_null[OF sets] u by auto |
1573 |
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
|
1574 |
finally show ?thesis . |
41705 | 1575 |
qed |
1576 |
||
56996 | 1577 |
lemma AE_iff_nn_integral: |
1578 |
"{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>N M (indicator {x. \<not> P x}) = 0" |
|
1579 |
by (subst nn_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
1580 |
sets.sets_Collect_neg indicator_def[abs_def] measurable_If) |
50001
382bd3173584
add syntax and a.e.-rules for (conditional) probability on predicates
hoelzl
parents:
49800
diff
changeset
|
1581 |
|
59000 | 1582 |
lemma nn_integral_less: |
1583 |
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
|
1584 |
assumes f: "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) \<noteq> \<infinity>" |
|
1585 |
assumes ord: "AE x in M. f x \<le> g x" "\<not> (AE x in M. g x \<le> f x)" |
|
1586 |
shows "(\<integral>\<^sup>+x. f x \<partial>M) < (\<integral>\<^sup>+x. g x \<partial>M)" |
|
1587 |
proof - |
|
1588 |
have "0 < (\<integral>\<^sup>+x. g x - f x \<partial>M)" |
|
1589 |
proof (intro order_le_neq_trans nn_integral_nonneg notI) |
|
1590 |
assume "0 = (\<integral>\<^sup>+x. g x - f x \<partial>M)" |
|
1591 |
then have "AE x in M. g x - f x \<le> 0" |
|
1592 |
using nn_integral_0_iff_AE[of "\<lambda>x. g x - f x" M] by simp |
|
1593 |
with f(1) ord(1) have "AE x in M. g x \<le> f x" |
|
1594 |
by eventually_elim (auto simp: ereal_minus_le_iff) |
|
1595 |
with ord show False |
|
1596 |
by simp |
|
1597 |
qed |
|
1598 |
also have "\<dots> = (\<integral>\<^sup>+x. g x \<partial>M) - (\<integral>\<^sup>+x. f x \<partial>M)" |
|
1599 |
by (subst nn_integral_diff) (auto simp: f ord) |
|
1600 |
finally show ?thesis |
|
1601 |
by (simp add: ereal_less_minus_iff f nn_integral_nonneg) |
|
1602 |
qed |
|
1603 |
||
56996 | 1604 |
lemma nn_integral_const_If: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1605 |
"(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)" |
56996 | 1606 |
by (auto intro!: nn_integral_0_iff_AE[THEN iffD2]) |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1607 |
|
56996 | 1608 |
lemma nn_integral_subalgebra: |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1609 |
assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x" |
47694 | 1610 |
and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A" |
56996 | 1611 |
shows "integral\<^sup>N N f = integral\<^sup>N M f" |
39092 | 1612 |
proof - |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1613 |
have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1614 |
using N by (auto simp: measurable_def) |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1615 |
have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1616 |
using N by (auto simp add: eventually_ae_filter null_sets_def) |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1617 |
have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1618 |
using N by auto |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1619 |
from f show ?thesis |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1620 |
apply induct |
56996 | 1621 |
apply (simp_all add: nn_integral_add nn_integral_cmult nn_integral_monotone_convergence_SUP N) |
1622 |
apply (auto intro!: nn_integral_cong cong: nn_integral_cong simp: N(2)[symmetric]) |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1623 |
done |
39092 | 1624 |
qed |
1625 |
||
56996 | 1626 |
lemma nn_integral_nat_function: |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1627 |
fixes f :: "'a \<Rightarrow> nat" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1628 |
assumes "f \<in> measurable M (count_space UNIV)" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1629 |
shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})" |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1630 |
proof - |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1631 |
def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1632 |
with assms have [measurable]: "\<And>i. F i \<in> sets M" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1633 |
by auto |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1634 |
|
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1635 |
{ fix x assume "x \<in> space M" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1636 |
have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1637 |
using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1638 |
then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1639 |
unfolding sums_ereal . |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1640 |
moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1641 |
using `x \<in> space M` by (simp add: one_ereal_def F_def) |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1642 |
ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1643 |
by (simp add: sums_iff) } |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1644 |
then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)" |
56996 | 1645 |
by (simp cong: nn_integral_cong) |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1646 |
also have "\<dots> = (\<Sum>i. emeasure M (F i))" |
56996 | 1647 |
by (simp add: nn_integral_suminf) |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1648 |
finally show ?thesis |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1649 |
by (simp add: F_def) |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1650 |
qed |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1651 |
|
56994 | 1652 |
subsection {* Integral under concrete measures *} |
1653 |
||
60064
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1654 |
lemma nn_integral_empty: |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1655 |
assumes "space M = {}" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1656 |
shows "nn_integral M f = 0" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1657 |
proof - |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1658 |
have "(\<integral>\<^sup>+ x. f x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1659 |
by(rule nn_integral_cong)(simp add: assms) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1660 |
thus ?thesis by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1661 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1662 |
|
56994 | 1663 |
subsubsection {* Distributions *} |
47694 | 1664 |
|
56996 | 1665 |
lemma nn_integral_distr': |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1666 |
assumes T: "T \<in> measurable M M'" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1667 |
and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x" |
56996 | 1668 |
shows "integral\<^sup>N (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)" |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1669 |
using f |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1670 |
proof induct |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1671 |
case (cong f g) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1672 |
with T show ?case |
56996 | 1673 |
apply (subst nn_integral_cong[of _ f g]) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1674 |
apply simp |
56996 | 1675 |
apply (subst nn_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"]) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1676 |
apply (simp add: measurable_def Pi_iff) |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1677 |
apply simp |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1678 |
done |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1679 |
next |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1680 |
case (set A) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1681 |
then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x" |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1682 |
by (auto simp: indicator_def) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1683 |
from set T show ?case |
56996 | 1684 |
by (subst nn_integral_cong[OF eq]) |
1685 |
(auto simp add: emeasure_distr intro!: nn_integral_indicator[symmetric] measurable_sets) |
|
1686 |
qed (simp_all add: measurable_compose[OF T] T nn_integral_cmult nn_integral_add |
|
1687 |
nn_integral_monotone_convergence_SUP le_fun_def incseq_def) |
|
47694 | 1688 |
|
56996 | 1689 |
lemma nn_integral_distr: |
1690 |
"T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>N (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)" |
|
1691 |
by (subst (1 2) nn_integral_max_0[symmetric]) |
|
1692 |
(simp add: nn_integral_distr') |
|
35692 | 1693 |
|
56994 | 1694 |
subsubsection {* Counting space *} |
47694 | 1695 |
|
1696 |
lemma simple_function_count_space[simp]: |
|
1697 |
"simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)" |
|
1698 |
unfolding simple_function_def by simp |
|
1699 |
||
56996 | 1700 |
lemma nn_integral_count_space: |
47694 | 1701 |
assumes A: "finite {a\<in>A. 0 < f a}" |
56996 | 1702 |
shows "integral\<^sup>N (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
35582 | 1703 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1704 |
have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) = |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1705 |
(\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)" |
56996 | 1706 |
by (auto intro!: nn_integral_cong |
57418 | 1707 |
simp add: indicator_def if_distrib setsum.If_cases[OF A] max_def le_less) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1708 |
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)" |
56996 | 1709 |
by (subst nn_integral_setsum) |
47694 | 1710 |
(simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le) |
1711 |
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
|
57418 | 1712 |
by (auto intro!: setsum.cong simp: nn_integral_cmult_indicator one_ereal_def[symmetric]) |
56996 | 1713 |
finally show ?thesis by (simp add: nn_integral_max_0) |
47694 | 1714 |
qed |
1715 |
||
56996 | 1716 |
lemma nn_integral_count_space_finite: |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1717 |
"finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))" |
56996 | 1718 |
by (subst nn_integral_max_0[symmetric]) |
57418 | 1719 |
(auto intro!: setsum.mono_neutral_left simp: nn_integral_count_space less_le) |
47694 | 1720 |
|
59000 | 1721 |
lemma nn_integral_count_space': |
1722 |
assumes "finite A" "\<And>x. x \<in> B \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0" "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "A \<subseteq> B" |
|
1723 |
shows "(\<integral>\<^sup>+x. f x \<partial>count_space B) = (\<Sum>x\<in>A. f x)" |
|
1724 |
proof - |
|
1725 |
have "(\<integral>\<^sup>+x. f x \<partial>count_space B) = (\<Sum>a | a \<in> B \<and> 0 < f a. f a)" |
|
1726 |
using assms(2,3) |
|
1727 |
by (intro nn_integral_count_space finite_subset[OF _ `finite A`]) (auto simp: less_le) |
|
1728 |
also have "\<dots> = (\<Sum>a\<in>A. f a)" |
|
1729 |
using assms by (intro setsum.mono_neutral_cong_left) (auto simp: less_le) |
|
1730 |
finally show ?thesis . |
|
1731 |
qed |
|
1732 |
||
59011
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1733 |
lemma nn_integral_bij_count_space: |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1734 |
assumes g: "bij_betw g A B" |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1735 |
shows "(\<integral>\<^sup>+x. f (g x) \<partial>count_space A) = (\<integral>\<^sup>+x. f x \<partial>count_space B)" |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1736 |
using g[THEN bij_betw_imp_funcset] |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1737 |
by (subst distr_bij_count_space[OF g, symmetric]) |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1738 |
(auto intro!: nn_integral_distr[symmetric]) |
4902a2fec434
add reindex rules for distr and nn_integral on count_space
hoelzl
parents:
59002
diff
changeset
|
1739 |
|
59000 | 1740 |
lemma nn_integral_indicator_finite: |
1741 |
fixes f :: "'a \<Rightarrow> ereal" |
|
1742 |
assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" and [measurable]: "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M" |
|
1743 |
shows "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<Sum>x\<in>A. f x * emeasure M {x})" |
|
1744 |
proof - |
|
1745 |
from f have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>a\<in>A. f a * indicator {a} x) \<partial>M)" |
|
1746 |
by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="\<lambda>a. x * a" for x] setsum.If_cases) |
|
1747 |
also have "\<dots> = (\<Sum>a\<in>A. f a * emeasure M {a})" |
|
1748 |
using nn by (subst nn_integral_setsum) (auto simp: nn_integral_cmult_indicator) |
|
1749 |
finally show ?thesis . |
|
1750 |
qed |
|
1751 |
||
57025 | 1752 |
lemma nn_integral_count_space_nat: |
1753 |
fixes f :: "nat \<Rightarrow> ereal" |
|
1754 |
assumes nonneg: "\<And>i. 0 \<le> f i" |
|
1755 |
shows "(\<integral>\<^sup>+i. f i \<partial>count_space UNIV) = (\<Sum>i. f i)" |
|
1756 |
proof - |
|
1757 |
have "(\<integral>\<^sup>+i. f i \<partial>count_space UNIV) = |
|
1758 |
(\<integral>\<^sup>+i. (\<Sum>j. f j * indicator {j} i) \<partial>count_space UNIV)" |
|
1759 |
proof (intro nn_integral_cong) |
|
1760 |
fix i |
|
1761 |
have "f i = (\<Sum>j\<in>{i}. f j * indicator {j} i)" |
|
1762 |
by simp |
|
1763 |
also have "\<dots> = (\<Sum>j. f j * indicator {j} i)" |
|
1764 |
by (rule suminf_finite[symmetric]) auto |
|
1765 |
finally show "f i = (\<Sum>j. f j * indicator {j} i)" . |
|
1766 |
qed |
|
1767 |
also have "\<dots> = (\<Sum>j. (\<integral>\<^sup>+i. f j * indicator {j} i \<partial>count_space UNIV))" |
|
1768 |
by (rule nn_integral_suminf) (auto simp: nonneg) |
|
1769 |
also have "\<dots> = (\<Sum>j. f j)" |
|
1770 |
by (simp add: nonneg nn_integral_cmult_indicator one_ereal_def[symmetric]) |
|
1771 |
finally show ?thesis . |
|
1772 |
qed |
|
1773 |
||
59426
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1774 |
lemma nn_integral_count_space_nn_integral: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1775 |
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1776 |
assumes "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1777 |
shows "(\<integral>\<^sup>+x. \<integral>\<^sup>+i. f i x \<partial>count_space I \<partial>M) = (\<integral>\<^sup>+i. \<integral>\<^sup>+x. f i x \<partial>M \<partial>count_space I)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1778 |
proof cases |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1779 |
assume "finite I" then show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1780 |
by (simp add: nn_integral_count_space_finite nn_integral_nonneg max.absorb2 nn_integral_setsum |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1781 |
nn_integral_max_0) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1782 |
next |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1783 |
assume "infinite I" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1784 |
then have [simp]: "I \<noteq> {}" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1785 |
by auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1786 |
note * = bij_betw_from_nat_into[OF `countable I` `infinite I`] |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1787 |
have **: "\<And>f. (\<And>i. 0 \<le> f i) \<Longrightarrow> (\<integral>\<^sup>+i. f i \<partial>count_space I) = (\<Sum>n. f (from_nat_into I n))" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1788 |
by (simp add: nn_integral_bij_count_space[symmetric, OF *] nn_integral_count_space_nat) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1789 |
show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1790 |
apply (subst (2) nn_integral_max_0[symmetric]) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1791 |
apply (simp add: ** nn_integral_nonneg nn_integral_suminf from_nat_into) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1792 |
apply (simp add: nn_integral_max_0) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1793 |
done |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1794 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1795 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1796 |
lemma emeasure_UN_countable: |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1797 |
assumes sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I[simp]: "countable I" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1798 |
assumes disj: "disjoint_family_on X I" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1799 |
shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1800 |
proof - |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1801 |
have eq: "\<And>x. indicator (UNION I X) x = \<integral>\<^sup>+ i. indicator (X i) x \<partial>count_space I" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1802 |
proof cases |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1803 |
fix x assume x: "x \<in> UNION I X" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1804 |
then obtain j where j: "x \<in> X j" "j \<in> I" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1805 |
by auto |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1806 |
with disj have "\<And>i. i \<in> I \<Longrightarrow> indicator (X i) x = (indicator {j} i::ereal)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1807 |
by (auto simp: disjoint_family_on_def split: split_indicator) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1808 |
with x j show "?thesis x" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1809 |
by (simp cong: nn_integral_cong_simp) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1810 |
qed (auto simp: nn_integral_0_iff_AE) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1811 |
|
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1812 |
note sets.countable_UN'[unfolded subset_eq, measurable] |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1813 |
have "emeasure M (UNION I X) = (\<integral>\<^sup>+x. indicator (UNION I X) x \<partial>M)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1814 |
by simp |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1815 |
also have "\<dots> = (\<integral>\<^sup>+i. \<integral>\<^sup>+x. indicator (X i) x \<partial>M \<partial>count_space I)" |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1816 |
by (simp add: eq nn_integral_count_space_nn_integral) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1817 |
finally show ?thesis |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1818 |
by (simp cong: nn_integral_cong_simp) |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1819 |
qed |
6fca83e88417
integral of the product of count spaces equals the integral of the count space of the product type
hoelzl
parents:
59425
diff
changeset
|
1820 |
|
57025 | 1821 |
lemma emeasure_countable_singleton: |
1822 |
assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" and X: "countable X" |
|
1823 |
shows "emeasure M X = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space X)" |
|
1824 |
proof - |
|
1825 |
have "emeasure M (\<Union>i\<in>X. {i}) = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space X)" |
|
1826 |
using assms by (intro emeasure_UN_countable) (auto simp: disjoint_family_on_def) |
|
1827 |
also have "(\<Union>i\<in>X. {i}) = X" by auto |
|
1828 |
finally show ?thesis . |
|
1829 |
qed |
|
1830 |
||
1831 |
lemma measure_eqI_countable: |
|
1832 |
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and A: "countable A" |
|
1833 |
assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" |
|
1834 |
shows "M = N" |
|
1835 |
proof (rule measure_eqI) |
|
1836 |
fix X assume "X \<in> sets M" |
|
1837 |
then have X: "X \<subseteq> A" by auto |
|
1838 |
moreover with A have "countable X" by (auto dest: countable_subset) |
|
1839 |
ultimately have |
|
1840 |
"emeasure M X = (\<integral>\<^sup>+a. emeasure M {a} \<partial>count_space X)" |
|
1841 |
"emeasure N X = (\<integral>\<^sup>+a. emeasure N {a} \<partial>count_space X)" |
|
1842 |
by (auto intro!: emeasure_countable_singleton) |
|
1843 |
moreover have "(\<integral>\<^sup>+a. emeasure M {a} \<partial>count_space X) = (\<integral>\<^sup>+a. emeasure N {a} \<partial>count_space X)" |
|
1844 |
using X by (intro nn_integral_cong eq) auto |
|
1845 |
ultimately show "emeasure M X = emeasure N X" |
|
1846 |
by simp |
|
1847 |
qed simp |
|
1848 |
||
59000 | 1849 |
lemma measure_eqI_countable_AE: |
1850 |
assumes [simp]: "sets M = UNIV" "sets N = UNIV" |
|
1851 |
assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>" and [simp]: "countable \<Omega>" |
|
1852 |
assumes eq: "\<And>x. x \<in> \<Omega> \<Longrightarrow> emeasure M {x} = emeasure N {x}" |
|
1853 |
shows "M = N" |
|
1854 |
proof (rule measure_eqI) |
|
1855 |
fix A |
|
1856 |
have "emeasure N A = emeasure N {x\<in>\<Omega>. x \<in> A}" |
|
1857 |
using ae by (intro emeasure_eq_AE) auto |
|
1858 |
also have "\<dots> = (\<integral>\<^sup>+x. emeasure N {x} \<partial>count_space {x\<in>\<Omega>. x \<in> A})" |
|
1859 |
by (intro emeasure_countable_singleton) auto |
|
1860 |
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space {x\<in>\<Omega>. x \<in> A})" |
|
1861 |
by (intro nn_integral_cong eq[symmetric]) auto |
|
1862 |
also have "\<dots> = emeasure M {x\<in>\<Omega>. x \<in> A}" |
|
1863 |
by (intro emeasure_countable_singleton[symmetric]) auto |
|
1864 |
also have "\<dots> = emeasure M A" |
|
1865 |
using ae by (intro emeasure_eq_AE) auto |
|
1866 |
finally show "emeasure M A = emeasure N A" .. |
|
1867 |
qed simp |
|
1868 |
||
60064
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1869 |
lemma nn_integral_monotone_convergence_SUP_nat': |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1870 |
fixes f :: "'a \<Rightarrow> nat \<Rightarrow> ereal" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1871 |
assumes chain: "Complete_Partial_Order.chain op \<le> (f ` Y)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1872 |
and nonempty: "Y \<noteq> {}" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1873 |
and nonneg: "\<And>i n. i \<in> Y \<Longrightarrow> f i n \<ge> 0" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1874 |
shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space UNIV) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space UNIV))" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1875 |
(is "?lhs = ?rhs" is "integral\<^sup>N ?M _ = _") |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1876 |
proof (rule order_class.order.antisym) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1877 |
show "?rhs \<le> ?lhs" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1878 |
by (auto intro!: SUP_least SUP_upper nn_integral_mono) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1879 |
next |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1880 |
have "\<And>x. \<exists>g. incseq g \<and> range g \<subseteq> (\<lambda>i. f i x) ` Y \<and> (SUP i:Y. f i x) = (SUP i. g i)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1881 |
unfolding Sup_class.SUP_def by(rule Sup_countable_SUP[unfolded Sup_class.SUP_def])(simp add: nonempty) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1882 |
then obtain g where incseq: "\<And>x. incseq (g x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1883 |
and range: "\<And>x. range (g x) \<subseteq> (\<lambda>i. f i x) ` Y" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1884 |
and sup: "\<And>x. (SUP i:Y. f i x) = (SUP i. g x i)" by moura |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1885 |
from incseq have incseq': "incseq (\<lambda>i x. g x i)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1886 |
by(blast intro: incseq_SucI le_funI dest: incseq_SucD) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1887 |
|
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1888 |
have "?lhs = \<integral>\<^sup>+ x. (SUP i. g x i) \<partial>?M" by(simp add: sup) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1889 |
also have "\<dots> = (SUP i. \<integral>\<^sup>+ x. g x i \<partial>?M)" using incseq' |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1890 |
by(rule nn_integral_monotone_convergence_SUP) simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1891 |
also have "\<dots> \<le> (SUP i:Y. \<integral>\<^sup>+ x. f i x \<partial>?M)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1892 |
proof(rule SUP_least) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1893 |
fix n |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1894 |
have "\<And>x. \<exists>i. g x n = f i x \<and> i \<in> Y" using range by blast |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1895 |
then obtain I where I: "\<And>x. g x n = f (I x) x" "\<And>x. I x \<in> Y" by moura |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1896 |
|
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1897 |
{ fix x |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1898 |
from range[of x] obtain i where "i \<in> Y" "g x n = f i x" by auto |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1899 |
hence "g x n \<ge> 0" using nonneg[of i x] by simp } |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1900 |
note nonneg_g = this |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1901 |
then have "(\<integral>\<^sup>+ x. g x n \<partial>count_space UNIV) = (\<Sum>x. g x n)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1902 |
by(rule nn_integral_count_space_nat) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1903 |
also have "\<dots> = (SUP m. \<Sum>x<m. g x n)" using nonneg_g |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1904 |
by(rule suminf_ereal_eq_SUP) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1905 |
also have "\<dots> \<le> (SUP i:Y. \<integral>\<^sup>+ x. f i x \<partial>?M)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1906 |
proof(rule SUP_mono) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1907 |
fix m |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1908 |
show "\<exists>m'\<in>Y. (\<Sum>x<m. g x n) \<le> (\<integral>\<^sup>+ x. f m' x \<partial>?M)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1909 |
proof(cases "m > 0") |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1910 |
case False |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1911 |
thus ?thesis using nonempty by(auto simp add: nn_integral_nonneg) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1912 |
next |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1913 |
case True |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1914 |
let ?Y = "I ` {..<m}" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1915 |
have "f ` ?Y \<subseteq> f ` Y" using I by auto |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1916 |
with chain have chain': "Complete_Partial_Order.chain op \<le> (f ` ?Y)" by(rule chain_subset) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1917 |
hence "Sup (f ` ?Y) \<in> f ` ?Y" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1918 |
by(rule ccpo_class.in_chain_finite)(auto simp add: True lessThan_empty_iff) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1919 |
then obtain m' where "m' < m" and m': "(SUP i:?Y. f i) = f (I m')" by auto |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1920 |
have "I m' \<in> Y" using I by blast |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1921 |
have "(\<Sum>x<m. g x n) \<le> (\<Sum>x<m. f (I m') x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1922 |
proof(rule setsum_mono) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1923 |
fix x |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1924 |
assume "x \<in> {..<m}" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1925 |
hence "x < m" by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1926 |
have "g x n = f (I x) x" by(simp add: I) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1927 |
also have "\<dots> \<le> (SUP i:?Y. f i) x" unfolding SUP_def Sup_fun_def image_image |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1928 |
using \<open>x \<in> {..<m}\<close> by(rule Sup_upper[OF imageI]) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1929 |
also have "\<dots> = f (I m') x" unfolding m' by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1930 |
finally show "g x n \<le> f (I m') x" . |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1931 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1932 |
also have "\<dots> \<le> (SUP m. (\<Sum>x<m. f (I m') x))" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1933 |
by(rule SUP_upper) simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1934 |
also have "\<dots> = (\<Sum>x. f (I m') x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1935 |
by(rule suminf_ereal_eq_SUP[symmetric])(simp add: nonneg \<open>I m' \<in> Y\<close>) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1936 |
also have "\<dots> = (\<integral>\<^sup>+ x. f (I m') x \<partial>?M)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1937 |
by(rule nn_integral_count_space_nat[symmetric])(simp add: nonneg \<open>I m' \<in> Y\<close>) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1938 |
finally show ?thesis using \<open>I m' \<in> Y\<close> by blast |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1939 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1940 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1941 |
finally show "(\<integral>\<^sup>+ x. g x n \<partial>count_space UNIV) \<le> \<dots>" . |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1942 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1943 |
finally show "?lhs \<le> ?rhs" . |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1944 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1945 |
|
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1946 |
lemma nn_integral_monotone_convergence_SUP_nat: |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1947 |
fixes f :: "'a \<Rightarrow> nat \<Rightarrow> ereal" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1948 |
assumes nonempty: "Y \<noteq> {}" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1949 |
and chain: "Complete_Partial_Order.chain op \<le> (f ` Y)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1950 |
shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space UNIV) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space UNIV))" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1951 |
(is "?lhs = ?rhs") |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1952 |
proof - |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1953 |
let ?f = "\<lambda>i x. max 0 (f i x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1954 |
have chain': "Complete_Partial_Order.chain op \<le> (?f ` Y)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1955 |
proof(rule chainI) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1956 |
fix g h |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1957 |
assume "g \<in> ?f ` Y" "h \<in> ?f ` Y" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1958 |
then obtain g' h' where gh: "g' \<in> Y" "h' \<in> Y" "g = ?f g'" "h = ?f h'" by blast |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1959 |
hence "f g' \<in> f ` Y" "f h' \<in> f ` Y" by blast+ |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1960 |
with chain have "f g' \<le> f h' \<or> f h' \<le> f g'" by(rule chainD) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1961 |
thus "g \<le> h \<or> h \<le> g" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1962 |
proof |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1963 |
assume "f g' \<le> f h'" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1964 |
hence "g \<le> h" using gh order_trans by(auto simp add: le_fun_def max_def) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1965 |
thus ?thesis .. |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1966 |
next |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1967 |
assume "f h' \<le> f g'" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1968 |
hence "h \<le> g" using gh order_trans by(auto simp add: le_fun_def max_def) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1969 |
thus ?thesis .. |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1970 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1971 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1972 |
have "?lhs = (\<integral>\<^sup>+ x. max 0 (SUP i:Y. f i x) \<partial>count_space UNIV)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1973 |
by(simp add: nn_integral_max_0) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1974 |
also have "\<dots> = (\<integral>\<^sup>+ x. (SUP i:Y. ?f i x) \<partial>count_space UNIV)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1975 |
proof(rule nn_integral_cong) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1976 |
fix x |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1977 |
have "max 0 (SUP i:Y. f i x) \<le> (SUP i:Y. ?f i x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1978 |
proof(cases "0 \<le> (SUP i:Y. f i x)") |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1979 |
case True |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1980 |
have "(SUP i:Y. f i x) \<le> (SUP i:Y. ?f i x)" by(rule SUP_mono)(auto intro: rev_bexI) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1981 |
with True show ?thesis by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1982 |
next |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1983 |
case False |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1984 |
have "0 \<le> (SUP i:Y. ?f i x)" using nonempty by(auto intro: SUP_upper2) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1985 |
thus ?thesis using False by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1986 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1987 |
moreover have "\<dots> \<le> max 0 (SUP i:Y. f i x)" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1988 |
proof(cases "(SUP i:Y. f i x) \<ge> 0") |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1989 |
case True |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1990 |
show ?thesis |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1991 |
by(rule SUP_least)(auto simp add: True max_def intro: SUP_upper) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1992 |
next |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1993 |
case False |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1994 |
hence "(SUP i:Y. f i x) \<le> 0" by simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1995 |
hence less: "\<forall>i\<in>Y. f i x \<le> 0" by(simp add: SUP_le_iff) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1996 |
show ?thesis by(rule SUP_least)(auto simp add: max_def less intro: SUP_upper) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1997 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1998 |
ultimately show "\<dots> = (SUP i:Y. ?f i x)" by(rule order.antisym) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
1999 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2000 |
also have "\<dots> = (SUP i:Y. (\<integral>\<^sup>+ x. ?f i x \<partial>count_space UNIV))" |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2001 |
using chain' nonempty by(rule nn_integral_monotone_convergence_SUP_nat') simp |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2002 |
also have "\<dots> = ?rhs" by(simp add: nn_integral_max_0) |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2003 |
finally show ?thesis . |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2004 |
qed |
63124d48a2ee
add lemma about monotone convergence for countable integrals over arbitrary sequences
Andreas Lochbihler
parents:
59779
diff
changeset
|
2005 |
|
56994 | 2006 |
subsubsection {* Measures with Restricted Space *} |
54417 | 2007 |
|
57137 | 2008 |
lemma simple_function_iff_borel_measurable: |
2009 |
fixes f :: "'a \<Rightarrow> 'x::{t2_space}" |
|
2010 |
shows "simple_function M f \<longleftrightarrow> finite (f ` space M) \<and> f \<in> borel_measurable M" |
|
2011 |
by (metis borel_measurable_simple_function simple_functionD(1) simple_function_borel_measurable) |
|
2012 |
||
2013 |
lemma simple_function_restrict_space_ereal: |
|
2014 |
fixes f :: "'a \<Rightarrow> ereal" |
|
2015 |
assumes "\<Omega> \<inter> space M \<in> sets M" |
|
2016 |
shows "simple_function (restrict_space M \<Omega>) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator \<Omega> x)" |
|
2017 |
proof - |
|
2018 |
{ assume "finite (f ` space (restrict_space M \<Omega>))" |
|
2019 |
then have "finite (f ` space (restrict_space M \<Omega>) \<union> {0})" by simp |
|
2020 |
then have "finite ((\<lambda>x. f x * indicator \<Omega> x) ` space M)" |
|
2021 |
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } |
|
2022 |
moreover |
|
2023 |
{ assume "finite ((\<lambda>x. f x * indicator \<Omega> x) ` space M)" |
|
2024 |
then have "finite (f ` space (restrict_space M \<Omega>))" |
|
2025 |
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } |
|
2026 |
ultimately show ?thesis |
|
2027 |
unfolding simple_function_iff_borel_measurable |
|
2028 |
borel_measurable_restrict_space_iff_ereal[OF assms] |
|
2029 |
by auto |
|
2030 |
qed |
|
2031 |
||
2032 |
lemma simple_function_restrict_space: |
|
2033 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
|
2034 |
assumes "\<Omega> \<inter> space M \<in> sets M" |
|
2035 |
shows "simple_function (restrict_space M \<Omega>) f \<longleftrightarrow> simple_function M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)" |
|
2036 |
proof - |
|
2037 |
{ assume "finite (f ` space (restrict_space M \<Omega>))" |
|
2038 |
then have "finite (f ` space (restrict_space M \<Omega>) \<union> {0})" by simp |
|
2039 |
then have "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x) ` space M)" |
|
2040 |
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } |
|
2041 |
moreover |
|
2042 |
{ assume "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x) ` space M)" |
|
2043 |
then have "finite (f ` space (restrict_space M \<Omega>))" |
|
2044 |
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) } |
|
2045 |
ultimately show ?thesis |
|
2046 |
unfolding simple_function_iff_borel_measurable |
|
2047 |
borel_measurable_restrict_space_iff[OF assms] |
|
2048 |
by auto |
|
2049 |
qed |
|
2050 |
||
2051 |
||
2052 |
lemma split_indicator_asm: "P (indicator Q x) = (\<not> (x \<in> Q \<and> \<not> P 1 \<or> x \<notin> Q \<and> \<not> P 0))" |
|
2053 |
by (auto split: split_indicator) |
|
2054 |
||
2055 |
lemma simple_integral_restrict_space: |
|
2056 |
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M" "simple_function (restrict_space M \<Omega>) f" |
|
2057 |
shows "simple_integral (restrict_space M \<Omega>) f = simple_integral M (\<lambda>x. f x * indicator \<Omega> x)" |
|
2058 |
using simple_function_restrict_space_ereal[THEN iffD1, OF \<Omega>, THEN simple_functionD(1)] |
|
2059 |
by (auto simp add: space_restrict_space emeasure_restrict_space[OF \<Omega>(1)] le_infI2 simple_integral_def |
|
2060 |
split: split_indicator split_indicator_asm |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
2061 |
intro!: setsum.mono_neutral_cong_left ereal_right_mult_cong[OF refl] arg_cong2[where f=emeasure]) |
57137 | 2062 |
|
2063 |
lemma one_not_less_zero[simp]: "\<not> 1 < (0::ereal)" |
|
2064 |
by (simp add: zero_ereal_def one_ereal_def) |
|
2065 |
||
56996 | 2066 |
lemma nn_integral_restrict_space: |
57137 | 2067 |
assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M" |
2068 |
shows "nn_integral (restrict_space M \<Omega>) f = nn_integral M (\<lambda>x. f x * indicator \<Omega> x)" |
|
2069 |
proof - |
|
2070 |
let ?R = "restrict_space M \<Omega>" and ?X = "\<lambda>M f. {s. simple_function M s \<and> s \<le> max 0 \<circ> f \<and> range s \<subseteq> {0 ..< \<infinity>}}" |
|
2071 |
have "integral\<^sup>S ?R ` ?X ?R f = integral\<^sup>S M ` ?X M (\<lambda>x. f x * indicator \<Omega> x)" |
|
2072 |
proof (safe intro!: image_eqI) |
|
2073 |
fix s assume s: "simple_function ?R s" "s \<le> max 0 \<circ> f" "range s \<subseteq> {0..<\<infinity>}" |
|
2074 |
from s show "integral\<^sup>S (restrict_space M \<Omega>) s = integral\<^sup>S M (\<lambda>x. s x * indicator \<Omega> x)" |
|
2075 |
by (intro simple_integral_restrict_space) auto |
|
2076 |
from s show "simple_function M (\<lambda>x. s x * indicator \<Omega> x)" |
|
2077 |
by (simp add: simple_function_restrict_space_ereal) |
|
2078 |
from s show "(\<lambda>x. s x * indicator \<Omega> x) \<le> max 0 \<circ> (\<lambda>x. f x * indicator \<Omega> x)" |
|
2079 |
"\<And>x. s x * indicator \<Omega> x \<in> {0..<\<infinity>}" |
|
2080 |
by (auto split: split_indicator simp: le_fun_def image_subset_iff) |
|
2081 |
next |
|
2082 |
fix s assume s: "simple_function M s" "s \<le> max 0 \<circ> (\<lambda>x. f x * indicator \<Omega> x)" "range s \<subseteq> {0..<\<infinity>}" |
|
2083 |
then have "simple_function M (\<lambda>x. s x * indicator (\<Omega> \<inter> space M) x)" (is ?s') |
|
2084 |
by (intro simple_function_mult simple_function_indicator) auto |
|
2085 |
also have "?s' \<longleftrightarrow> simple_function M (\<lambda>x. s x * indicator \<Omega> x)" |
|
2086 |
by (rule simple_function_cong) (auto split: split_indicator) |
|
2087 |
finally show sf: "simple_function (restrict_space M \<Omega>) s" |
|
2088 |
by (simp add: simple_function_restrict_space_ereal) |
|
2089 |
||
2090 |
from s have s_eq: "s = (\<lambda>x. s x * indicator \<Omega> x)" |
|
2091 |
by (auto simp add: fun_eq_iff le_fun_def image_subset_iff |
|
2092 |
split: split_indicator split_indicator_asm |
|
2093 |
intro: antisym) |
|
2094 |
||
2095 |
show "integral\<^sup>S M s = integral\<^sup>S (restrict_space M \<Omega>) s" |
|
2096 |
by (subst s_eq) (rule simple_integral_restrict_space[symmetric, OF \<Omega> sf]) |
|
2097 |
show "\<And>x. s x \<in> {0..<\<infinity>}" |
|
2098 |
using s by (auto simp: image_subset_iff) |
|
2099 |
from s show "s \<le> max 0 \<circ> f" |
|
2100 |
by (subst s_eq) (auto simp: image_subset_iff le_fun_def split: split_indicator split_indicator_asm) |
|
2101 |
qed |
|
2102 |
then show ?thesis |
|
2103 |
unfolding nn_integral_def_finite SUP_def by simp |
|
54417 | 2104 |
qed |
2105 |
||
59000 | 2106 |
lemma nn_integral_count_space_indicator: |
59779 | 2107 |
assumes "NO_MATCH (UNIV::'a set) (X::'a set)" |
59000 | 2108 |
shows "(\<integral>\<^sup>+x. f x \<partial>count_space X) = (\<integral>\<^sup>+x. f x * indicator X x \<partial>count_space UNIV)" |
2109 |
by (simp add: nn_integral_restrict_space[symmetric] restrict_count_space) |
|
2110 |
||
59425 | 2111 |
lemma nn_integral_count_space_eq: |
2112 |
"(\<And>x. x \<in> A - B \<Longrightarrow> f x = 0) \<Longrightarrow> (\<And>x. x \<in> B - A \<Longrightarrow> f x = 0) \<Longrightarrow> |
|
2113 |
(\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<integral>\<^sup>+x. f x \<partial>count_space B)" |
|
2114 |
by (auto simp: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator) |
|
2115 |
||
59023 | 2116 |
lemma nn_integral_ge_point: |
2117 |
assumes "x \<in> A" |
|
2118 |
shows "p x \<le> \<integral>\<^sup>+ x. p x \<partial>count_space A" |
|
2119 |
proof - |
|
2120 |
from assms have "p x \<le> \<integral>\<^sup>+ x. p x \<partial>count_space {x}" |
|
2121 |
by(auto simp add: nn_integral_count_space_finite max_def) |
|
2122 |
also have "\<dots> = \<integral>\<^sup>+ x'. p x' * indicator {x} x' \<partial>count_space A" |
|
2123 |
using assms by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong) |
|
2124 |
also have "\<dots> \<le> \<integral>\<^sup>+ x. max 0 (p x) \<partial>count_space A" |
|
2125 |
by(rule nn_integral_mono)(simp add: indicator_def) |
|
2126 |
also have "\<dots> = \<integral>\<^sup>+ x. p x \<partial>count_space A" by(simp add: nn_integral_def o_def) |
|
2127 |
finally show ?thesis . |
|
2128 |
qed |
|
2129 |
||
56994 | 2130 |
subsubsection {* Measure spaces with an associated density *} |
47694 | 2131 |
|
2132 |
definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2133 |
"density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)" |
35582 | 2134 |
|
47694 | 2135 |
lemma |
59048 | 2136 |
shows sets_density[simp, measurable_cong]: "sets (density M f) = sets M" |
47694 | 2137 |
and space_density[simp]: "space (density M f) = space M" |
2138 |
by (auto simp: density_def) |
|
2139 |
||
50003 | 2140 |
(* FIXME: add conversion to simplify space, sets and measurable *) |
2141 |
lemma space_density_imp[measurable_dest]: |
|
2142 |
"\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto |
|
2143 |
||
47694 | 2144 |
lemma |
2145 |
shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'" |
|
2146 |
and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'" |
|
2147 |
and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u" |
|
2148 |
unfolding measurable_def simple_function_def by simp_all |
|
2149 |
||
2150 |
lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow> |
|
2151 |
(AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'" |
|
56996 | 2152 |
unfolding density_def by (auto intro!: measure_of_eq nn_integral_cong_AE sets.space_closed) |
47694 | 2153 |
|
2154 |
lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))" |
|
2155 |
proof - |
|
2156 |
have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)" |
|
2157 |
by (auto simp: indicator_def) |
|
2158 |
then show ?thesis |
|
56996 | 2159 |
unfolding density_def by (simp add: nn_integral_max_0) |
47694 | 2160 |
qed |
2161 |
||
2162 |
lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))" |
|
2163 |
by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max) |
|
38656 | 2164 |
|
47694 | 2165 |
lemma emeasure_density: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
2166 |
assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2167 |
shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)" |
47694 | 2168 |
(is "_ = ?\<mu> A") |
2169 |
unfolding density_def |
|
2170 |
proof (rule emeasure_measure_of_sigma) |
|
2171 |
show "sigma_algebra (space M) (sets M)" .. |
|
2172 |
show "positive (sets M) ?\<mu>" |
|
56996 | 2173 |
using f by (auto simp: positive_def intro!: nn_integral_nonneg) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2174 |
have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'") |
56996 | 2175 |
apply (subst nn_integral_max_0[symmetric]) |
2176 |
apply (intro ext nn_integral_cong_AE AE_I2) |
|
47694 | 2177 |
apply (auto simp: indicator_def) |
2178 |
done |
|
2179 |
show "countably_additive (sets M) ?\<mu>" |
|
2180 |
unfolding \<mu>_eq |
|
2181 |
proof (intro countably_additiveI) |
|
2182 |
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
2183 |
then have "\<And>i. A i \<in> sets M" by auto |
47694 | 2184 |
then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M" |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
2185 |
by (auto simp: set_eq_iff) |
47694 | 2186 |
assume disj: "disjoint_family A" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2187 |
have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)" |
56996 | 2188 |
using f * by (simp add: nn_integral_suminf) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2189 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f |
56996 | 2190 |
by (auto intro!: suminf_cmult_ereal nn_integral_cong_AE) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2191 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)" |
47694 | 2192 |
unfolding suminf_indicator[OF disj] .. |
2193 |
finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp |
|
2194 |
qed |
|
2195 |
qed fact |
|
38656 | 2196 |
|
47694 | 2197 |
lemma null_sets_density_iff: |
2198 |
assumes f: "f \<in> borel_measurable M" |
|
2199 |
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)" |
|
2200 |
proof - |
|
2201 |
{ assume "A \<in> sets M" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2202 |
have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)" |
56996 | 2203 |
apply (subst nn_integral_max_0[symmetric]) |
2204 |
apply (intro nn_integral_cong) |
|
47694 | 2205 |
apply (auto simp: indicator_def) |
2206 |
done |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2207 |
have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> |
47694 | 2208 |
emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0" |
2209 |
unfolding eq |
|
2210 |
using f `A \<in> sets M` |
|
56996 | 2211 |
by (intro nn_integral_0_iff) auto |
47694 | 2212 |
also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)" |
2213 |
using f `A \<in> sets M` |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
2214 |
by (intro AE_iff_measurable[OF _ refl, symmetric]) auto |
47694 | 2215 |
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)" |
2216 |
by (auto simp add: indicator_def max_def split: split_if_asm) |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2217 |
finally have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . } |
47694 | 2218 |
with f show ?thesis |
2219 |
by (simp add: null_sets_def emeasure_density cong: conj_cong) |
|
2220 |
qed |
|
2221 |
||
2222 |
lemma AE_density: |
|
2223 |
assumes f: "f \<in> borel_measurable M" |
|
2224 |
shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)" |
|
2225 |
proof |
|
2226 |
assume "AE x in density M f. P x" |
|
2227 |
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" |
|
2228 |
by (auto simp: eventually_ae_filter null_sets_density_iff) |
|
2229 |
then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto |
|
2230 |
with ae show "AE x in M. 0 < f x \<longrightarrow> P x" |
|
2231 |
by (rule eventually_elim2) auto |
|
2232 |
next |
|
2233 |
fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x" |
|
2234 |
then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M" |
|
2235 |
by (auto simp: eventually_ae_filter) |
|
2236 |
then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}" |
|
2237 |
"N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
2238 |
using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in) |
47694 | 2239 |
show "AE x in density M f. P x" |
2240 |
using ae2 |
|
2241 |
unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f] |
|
2242 |
by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *) |
|
2243 |
(auto elim: eventually_elim2) |
|
35582 | 2244 |
qed |
2245 |
||
56996 | 2246 |
lemma nn_integral_density': |
47694 | 2247 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
2248 |
assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x" |
56996 | 2249 |
shows "integral\<^sup>N (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)" |
49798 | 2250 |
using g proof induct |
2251 |
case (cong u v) |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
2252 |
then show ?case |
56996 | 2253 |
apply (subst nn_integral_cong[OF cong(3)]) |
2254 |
apply (simp_all cong: nn_integral_cong) |
|
49798 | 2255 |
done |
2256 |
next |
|
2257 |
case (set A) then show ?case |
|
2258 |
by (simp add: emeasure_density f) |
|
2259 |
next |
|
2260 |
case (mult u c) |
|
2261 |
moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps) |
|
2262 |
ultimately show ?case |
|
56996 | 2263 |
using f by (simp add: nn_integral_cmult) |
49798 | 2264 |
next |
2265 |
case (add u v) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
2266 |
then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x" |
49798 | 2267 |
by (simp add: ereal_right_distrib) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
2268 |
with add f show ?case |
56996 | 2269 |
by (auto simp add: nn_integral_add ereal_zero_le_0_iff intro!: nn_integral_add[symmetric]) |
49798 | 2270 |
next |
2271 |
case (seq U) |
|
2272 |
from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59426
diff
changeset
|
2273 |
by eventually_elim (simp add: SUP_ereal_mult_left seq) |
49798 | 2274 |
from seq f show ?case |
56996 | 2275 |
apply (simp add: nn_integral_monotone_convergence_SUP) |
2276 |
apply (subst nn_integral_cong_AE[OF eq]) |
|
2277 |
apply (subst nn_integral_monotone_convergence_SUP_AE) |
|
49798 | 2278 |
apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono) |
2279 |
done |
|
47694 | 2280 |
qed |
38705 | 2281 |
|
56996 | 2282 |
lemma nn_integral_density: |
49798 | 2283 |
"f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> |
56996 | 2284 |
integral\<^sup>N (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)" |
2285 |
by (subst (1 2) nn_integral_max_0[symmetric]) |
|
2286 |
(auto intro!: nn_integral_cong_AE |
|
2287 |
simp: measurable_If max_def ereal_zero_le_0_iff nn_integral_density') |
|
49798 | 2288 |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2289 |
lemma density_distr: |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2290 |
assumes [measurable]: "f \<in> borel_measurable N" "X \<in> measurable M N" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2291 |
shows "density (distr M N X) f = distr (density M (\<lambda>x. f (X x))) N X" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2292 |
by (intro measure_eqI) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2293 |
(auto simp add: emeasure_density nn_integral_distr emeasure_distr |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2294 |
split: split_indicator intro!: nn_integral_cong) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57137
diff
changeset
|
2295 |
|
47694 | 2296 |
lemma emeasure_restricted: |
2297 |
assumes S: "S \<in> sets M" and X: "X \<in> sets M" |
|
2298 |
shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)" |
|
38705 | 2299 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2300 |
have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)" |
47694 | 2301 |
using S X by (simp add: emeasure_density) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2302 |
also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)" |
56996 | 2303 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
47694 | 2304 |
also have "\<dots> = emeasure M (S \<inter> X)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
2305 |
using S X by (simp add: sets.Int) |
47694 | 2306 |
finally show ?thesis . |
2307 |
qed |
|
2308 |
||
2309 |
lemma measure_restricted: |
|
2310 |
"S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)" |
|
2311 |
by (simp add: emeasure_restricted measure_def) |
|
2312 |
||
2313 |
lemma (in finite_measure) finite_measure_restricted: |
|
2314 |
"S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))" |
|
2315 |
by default (simp add: emeasure_restricted) |
|
2316 |
||
2317 |
lemma emeasure_density_const: |
|
2318 |
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A" |
|
56996 | 2319 |
by (auto simp: nn_integral_cmult_indicator emeasure_density) |
47694 | 2320 |
|
2321 |
lemma measure_density_const: |
|
59000 | 2322 |
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A" |
47694 | 2323 |
by (auto simp: emeasure_density_const measure_def) |
2324 |
||
2325 |
lemma density_density_eq: |
|
2326 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> |
|
2327 |
density (density M f) g = density M (\<lambda>x. f x * g x)" |
|
56996 | 2328 |
by (auto intro!: measure_eqI simp: emeasure_density nn_integral_density ac_simps) |
47694 | 2329 |
|
2330 |
lemma distr_density_distr: |
|
2331 |
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
2332 |
and inv: "\<forall>x\<in>space M. T' (T x) = x" |
|
2333 |
assumes f: "f \<in> borel_measurable M'" |
|
2334 |
shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L") |
|
2335 |
proof (rule measure_eqI) |
|
2336 |
fix A assume A: "A \<in> sets ?R" |
|
2337 |
{ fix x assume "x \<in> space M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
2338 |
with sets.sets_into_space[OF A] |
47694 | 2339 |
have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)" |
2340 |
using T inv by (auto simp: indicator_def measurable_space) } |
|
2341 |
with A T T' f show "emeasure ?R A = emeasure ?L A" |
|
2342 |
by (simp add: measurable_comp emeasure_density emeasure_distr |
|
56996 | 2343 |
nn_integral_distr measurable_sets cong: nn_integral_cong) |
47694 | 2344 |
qed simp |
2345 |
||
2346 |
lemma density_density_divide: |
|
2347 |
fixes f g :: "'a \<Rightarrow> real" |
|
2348 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
2349 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
2350 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
2351 |
shows "density (density M f) (\<lambda>x. g x / f x) = density M g" |
|
2352 |
proof - |
|
2353 |
have "density M g = density M (\<lambda>x. f x * (g x / f x))" |
|
2354 |
using f g ac by (auto intro!: density_cong measurable_If) |
|
2355 |
then show ?thesis |
|
2356 |
using f g by (subst density_density_eq) auto |
|
38705 | 2357 |
qed |
2358 |
||
59425 | 2359 |
lemma density_1: "density M (\<lambda>_. 1) = M" |
2360 |
by (intro measure_eqI) (auto simp: emeasure_density) |
|
2361 |
||
2362 |
lemma emeasure_density_add: |
|
2363 |
assumes X: "X \<in> sets M" |
|
2364 |
assumes Mf[measurable]: "f \<in> borel_measurable M" |
|
2365 |
assumes Mg[measurable]: "g \<in> borel_measurable M" |
|
2366 |
assumes nonnegf: "\<And>x. x \<in> space M \<Longrightarrow> f x \<ge> 0" |
|
2367 |
assumes nonnegg: "\<And>x. x \<in> space M \<Longrightarrow> g x \<ge> 0" |
|
2368 |
shows "emeasure (density M f) X + emeasure (density M g) X = |
|
2369 |
emeasure (density M (\<lambda>x. f x + g x)) X" |
|
2370 |
using assms |
|
2371 |
apply (subst (1 2 3) emeasure_density, simp_all) [] |
|
2372 |
apply (subst nn_integral_add[symmetric], simp_all) [] |
|
2373 |
apply (intro nn_integral_cong, simp split: split_indicator) |
|
2374 |
done |
|
2375 |
||
56994 | 2376 |
subsubsection {* Point measure *} |
47694 | 2377 |
|
2378 |
definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
2379 |
"point_measure A f = density (count_space A) f" |
|
2380 |
||
2381 |
lemma |
|
2382 |
shows space_point_measure: "space (point_measure A f) = A" |
|
2383 |
and sets_point_measure: "sets (point_measure A f) = Pow A" |
|
2384 |
by (auto simp: point_measure_def) |
|
2385 |
||
59048 | 2386 |
lemma sets_point_measure_count_space[measurable_cong]: "sets (point_measure A f) = sets (count_space A)" |
2387 |
by (simp add: sets_point_measure) |
|
2388 |
||
47694 | 2389 |
lemma measurable_point_measure_eq1[simp]: |
2390 |
"g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M" |
|
2391 |
unfolding point_measure_def by simp |
|
2392 |
||
2393 |
lemma measurable_point_measure_eq2_finite[simp]: |
|
2394 |
"finite A \<Longrightarrow> |
|
2395 |
g \<in> measurable M (point_measure A f) \<longleftrightarrow> |
|
2396 |
(g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))" |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
2397 |
unfolding point_measure_def by (simp add: measurable_count_space_eq2) |
47694 | 2398 |
|
2399 |
lemma simple_function_point_measure[simp]: |
|
2400 |
"simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)" |
|
2401 |
by (simp add: point_measure_def) |
|
2402 |
||
2403 |
lemma emeasure_point_measure: |
|
2404 |
assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A" |
|
2405 |
shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)" |
|
35977 | 2406 |
proof - |
47694 | 2407 |
have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}" |
2408 |
using `X \<subseteq> A` by auto |
|
2409 |
with A show ?thesis |
|
56996 | 2410 |
by (simp add: emeasure_density nn_integral_count_space ereal_zero_le_0_iff |
47694 | 2411 |
point_measure_def indicator_def) |
35977 | 2412 |
qed |
2413 |
||
47694 | 2414 |
lemma emeasure_point_measure_finite: |
49795 | 2415 |
"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)" |
57418 | 2416 |
by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le) |
47694 | 2417 |
|
49795 | 2418 |
lemma emeasure_point_measure_finite2: |
2419 |
"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)" |
|
2420 |
by (subst emeasure_point_measure) |
|
57418 | 2421 |
(auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le) |
49795 | 2422 |
|
47694 | 2423 |
lemma null_sets_point_measure_iff: |
2424 |
"X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)" |
|
2425 |
by (auto simp: AE_count_space null_sets_density_iff point_measure_def) |
|
2426 |
||
2427 |
lemma AE_point_measure: |
|
2428 |
"(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)" |
|
2429 |
unfolding point_measure_def |
|
2430 |
by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def) |
|
2431 |
||
56996 | 2432 |
lemma nn_integral_point_measure: |
47694 | 2433 |
"finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow> |
56996 | 2434 |
integral\<^sup>N (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)" |
47694 | 2435 |
unfolding point_measure_def |
2436 |
apply (subst density_max_0) |
|
56996 | 2437 |
apply (subst nn_integral_density) |
2438 |
apply (simp_all add: AE_count_space nn_integral_density) |
|
2439 |
apply (subst nn_integral_count_space ) |
|
57418 | 2440 |
apply (auto intro!: setsum.cong simp: max_def ereal_zero_less_0_iff) |
47694 | 2441 |
apply (rule finite_subset) |
2442 |
prefer 2 |
|
2443 |
apply assumption |
|
2444 |
apply auto |
|
2445 |
done |
|
2446 |
||
56996 | 2447 |
lemma nn_integral_point_measure_finite: |
47694 | 2448 |
"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> |
56996 | 2449 |
integral\<^sup>N (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)" |
57418 | 2450 |
by (subst nn_integral_point_measure) (auto intro!: setsum.mono_neutral_left simp: less_le) |
47694 | 2451 |
|
56994 | 2452 |
subsubsection {* Uniform measure *} |
47694 | 2453 |
|
2454 |
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)" |
|
2455 |
||
2456 |
lemma |
|
59048 | 2457 |
shows sets_uniform_measure[simp, measurable_cong]: "sets (uniform_measure M A) = sets M" |
47694 | 2458 |
and space_uniform_measure[simp]: "space (uniform_measure M A) = space M" |
2459 |
by (auto simp: uniform_measure_def) |
|
2460 |
||
2461 |
lemma emeasure_uniform_measure[simp]: |
|
2462 |
assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
|
2463 |
shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A" |
|
2464 |
proof - |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2465 |
from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)" |
47694 | 2466 |
by (auto simp add: uniform_measure_def emeasure_density split: split_indicator |
56996 | 2467 |
intro!: nn_integral_cong) |
47694 | 2468 |
also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A" |
2469 |
using A B |
|
56996 | 2470 |
by (subst nn_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg) |
47694 | 2471 |
finally show ?thesis . |
2472 |
qed |
|
2473 |
||
2474 |
lemma measure_uniform_measure[simp]: |
|
2475 |
assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M" |
|
2476 |
shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A" |
|
2477 |
using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A |
|
2478 |
by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def) |
|
2479 |
||
58606 | 2480 |
lemma AE_uniform_measureI: |
2481 |
"A \<in> sets M \<Longrightarrow> (AE x in M. x \<in> A \<longrightarrow> P x) \<Longrightarrow> (AE x in uniform_measure M A. P x)" |
|
2482 |
unfolding uniform_measure_def by (auto simp: AE_density) |
|
2483 |
||
59000 | 2484 |
lemma emeasure_uniform_measure_1: |
2485 |
"emeasure M S \<noteq> 0 \<Longrightarrow> emeasure M S \<noteq> \<infinity> \<Longrightarrow> emeasure (uniform_measure M S) S = 1" |
|
2486 |
by (subst emeasure_uniform_measure) |
|
2487 |
(simp_all add: emeasure_nonneg emeasure_neq_0_sets) |
|
2488 |
||
2489 |
lemma nn_integral_uniform_measure: |
|
2490 |
assumes f[measurable]: "f \<in> borel_measurable M" and "\<And>x. 0 \<le> f x" and S[measurable]: "S \<in> sets M" |
|
2491 |
shows "(\<integral>\<^sup>+x. f x \<partial>uniform_measure M S) = (\<integral>\<^sup>+x. f x * indicator S x \<partial>M) / emeasure M S" |
|
2492 |
proof - |
|
2493 |
{ assume "emeasure M S = \<infinity>" |
|
2494 |
then have ?thesis |
|
2495 |
by (simp add: uniform_measure_def nn_integral_density f) } |
|
2496 |
moreover |
|
2497 |
{ assume [simp]: "emeasure M S = 0" |
|
2498 |
then have ae: "AE x in M. x \<notin> S" |
|
2499 |
using sets.sets_into_space[OF S] |
|
2500 |
by (subst AE_iff_measurable[OF _ refl]) (simp_all add: subset_eq cong: rev_conj_cong) |
|
2501 |
from ae have "(\<integral>\<^sup>+ x. indicator S x * f x / 0 \<partial>M) = 0" |
|
2502 |
by (subst nn_integral_0_iff_AE) auto |
|
2503 |
moreover from ae have "(\<integral>\<^sup>+ x. f x * indicator S x \<partial>M) = 0" |
|
2504 |
by (subst nn_integral_0_iff_AE) auto |
|
2505 |
ultimately have ?thesis |
|
2506 |
by (simp add: uniform_measure_def nn_integral_density f) } |
|
2507 |
moreover |
|
2508 |
{ assume "emeasure M S \<noteq> 0" "emeasure M S \<noteq> \<infinity>" |
|
2509 |
moreover then have "0 < emeasure M S" |
|
2510 |
by (simp add: emeasure_nonneg less_le) |
|
2511 |
ultimately have ?thesis |
|
2512 |
unfolding uniform_measure_def |
|
2513 |
apply (subst nn_integral_density) |
|
2514 |
apply (auto simp: f nn_integral_divide intro!: zero_le_divide_ereal) |
|
2515 |
apply (simp add: mult.commute) |
|
2516 |
done } |
|
2517 |
ultimately show ?thesis by blast |
|
2518 |
qed |
|
2519 |
||
2520 |
lemma AE_uniform_measure: |
|
2521 |
assumes "emeasure M A \<noteq> 0" "emeasure M A < \<infinity>" |
|
2522 |
shows "(AE x in uniform_measure M A. P x) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> P x)" |
|
2523 |
proof - |
|
2524 |
have "A \<in> sets M" |
|
2525 |
using `emeasure M A \<noteq> 0` by (metis emeasure_notin_sets) |
|
2526 |
moreover have "\<And>x. 0 < indicator A x / emeasure M A \<longleftrightarrow> x \<in> A" |
|
2527 |
using emeasure_nonneg[of M A] assms |
|
2528 |
by (cases "emeasure M A") (auto split: split_indicator simp: one_ereal_def) |
|
2529 |
ultimately show ?thesis |
|
2530 |
unfolding uniform_measure_def by (simp add: AE_density) |
|
2531 |
qed |
|
2532 |
||
59425 | 2533 |
subsubsection {* Null measure *} |
2534 |
||
2535 |
lemma null_measure_eq_density: "null_measure M = density M (\<lambda>_. 0)" |
|
2536 |
by (intro measure_eqI) (simp_all add: emeasure_density) |
|
2537 |
||
2538 |
lemma nn_integral_null_measure[simp]: "(\<integral>\<^sup>+x. f x \<partial>null_measure M) = 0" |
|
2539 |
by (auto simp add: nn_integral_def simple_integral_def SUP_constant bot_ereal_def le_fun_def |
|
2540 |
intro!: exI[of _ "\<lambda>x. 0"]) |
|
2541 |
||
2542 |
lemma density_null_measure[simp]: "density (null_measure M) f = null_measure M" |
|
2543 |
proof (intro measure_eqI) |
|
2544 |
fix A show "emeasure (density (null_measure M) f) A = emeasure (null_measure M) A" |
|
2545 |
by (simp add: density_def) (simp only: null_measure_def[symmetric] emeasure_null_measure) |
|
2546 |
qed simp |
|
2547 |
||
56994 | 2548 |
subsubsection {* Uniform count measure *} |
47694 | 2549 |
|
2550 |
definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)" |
|
2551 |
||
2552 |
lemma |
|
2553 |
shows space_uniform_count_measure: "space (uniform_count_measure A) = A" |
|
2554 |
and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A" |
|
2555 |
unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure) |
|
59048 | 2556 |
|
2557 |
lemma sets_uniform_count_measure_count_space[measurable_cong]: |
|
2558 |
"sets (uniform_count_measure A) = sets (count_space A)" |
|
2559 |
by (simp add: sets_uniform_count_measure) |
|
47694 | 2560 |
|
2561 |
lemma emeasure_uniform_count_measure: |
|
2562 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A" |
|
2563 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def) |
|
2564 |
||
2565 |
lemma measure_uniform_count_measure: |
|
2566 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A" |
|
2567 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def) |
|
2568 |
||
35748 | 2569 |
end |