author | hoelzl |
Mon, 19 May 2014 14:26:58 +0200 | |
changeset 56996 | 891e992e510f |
parent 56994 | 8d5e5ec1cac3 |
child 57025 | e7fd64f82876 |
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
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(* Title: HOL/Probability/Nonnegative_Lebesgue_Integration.thy |
42067 | 2 |
Author: Johannes Hölzl, TU München |
3 |
Author: Armin Heller, TU München |
|
4 |
*) |
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|
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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header {* Lebesgue Integration for Nonnegative Functions *} |
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|
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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theory Nonnegative_Lebesgue_Integration |
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imports Measure_Space Borel_Space |
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begin |
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||
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lemma indicator_less_ereal[simp]: |
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"indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)" |
|
14 |
by (simp add: indicator_def not_le) |
|
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subsection "Simple function" |
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text {* |
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Our simple functions are not restricted to nonnegative real numbers. Instead |
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they are just functions with a finite range and are measurable when singleton |
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sets are measurable. |
|
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|
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*} |
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definition "simple_function M g \<longleftrightarrow> |
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finite (g ` space M) \<and> |
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(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)" |
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lemma simple_functionD: |
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assumes "simple_function M g" |
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shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M" |
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proof - |
34 |
show "finite (g ` space M)" |
|
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using assms unfolding simple_function_def by auto |
|
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have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto |
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also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto |
|
38 |
finally show "g -` X \<inter> space M \<in> sets M" using assms |
|
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by (auto simp del: UN_simps simp: simple_function_def) |
40871 | 40 |
qed |
36624 | 41 |
|
56949 | 42 |
lemma measurable_simple_function[measurable_dest]: |
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"simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)" |
|
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unfolding simple_function_def measurable_def |
|
45 |
proof safe |
|
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fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M" |
|
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then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M" |
|
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by (intro sets.finite_UN) auto |
|
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also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M" |
|
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by (auto split: split_if_asm) |
|
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finally show "f -` A \<inter> space M \<in> sets M" . |
|
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qed simp |
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||
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lemma borel_measurable_simple_function: |
|
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"simple_function M f \<Longrightarrow> f \<in> borel_measurable M" |
|
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by (auto dest!: measurable_simple_function simp: measurable_def) |
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lemma simple_function_measurable2[intro]: |
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assumes "simple_function M f" "simple_function M g" |
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parents:
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shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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proof - |
cdf7693bbe08
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have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)" |
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by auto |
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reworked Probability theory: measures are not type restricted to positive extended reals
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then show ?thesis using assms[THEN simple_functionD(2)] by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
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qed |
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|
47694 | 67 |
lemma simple_function_indicator_representation: |
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fixes f ::"'a \<Rightarrow> ereal" |
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assumes f: "simple_function M f" and x: "x \<in> space M" |
38656 | 70 |
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)" |
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(is "?l = ?r") |
|
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proof - |
|
38705 | 73 |
have "?r = (\<Sum>y \<in> f ` space M. |
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(if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))" |
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by (auto intro!: setsum_cong2) |
|
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also have "... = f x * indicator (f -` {f x} \<inter> space M) x" |
|
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using assms by (auto dest: simple_functionD simp: setsum_delta) |
|
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also have "... = f x" using x by (auto simp: indicator_def) |
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finally show ?thesis by auto |
|
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qed |
|
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|
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lemma simple_function_notspace: |
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"simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h") |
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proof - |
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have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto |
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hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset) |
|
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have "?h -` {0} \<inter> space M = space M" by auto |
|
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thus ?thesis unfolding simple_function_def by auto |
|
89 |
qed |
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||
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lemma simple_function_cong: |
38656 | 92 |
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" |
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the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
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shows "simple_function M f \<longleftrightarrow> simple_function M g" |
38656 | 94 |
proof - |
95 |
have "f ` space M = g ` space M" |
|
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"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
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using assms by (auto intro!: image_eqI) |
|
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thus ?thesis unfolding simple_function_def using assms by simp |
|
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qed |
|
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lemma simple_function_cong_algebra: |
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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:
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|
103 |
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:
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|
104 |
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:
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|
105 |
|
47694 | 106 |
lemma simple_function_borel_measurable: |
41981
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107 |
fixes f :: "'a \<Rightarrow> 'x::{t2_space}" |
38656 | 108 |
assumes "f \<in> borel_measurable M" and "finite (f ` space M)" |
41689
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hoelzl
parents:
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|
109 |
shows "simple_function M f" |
38656 | 110 |
using assms unfolding simple_function_def |
111 |
by (auto intro: borel_measurable_vimage) |
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lemma simple_function_eq_measurable: |
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fixes f :: "'a \<Rightarrow> ereal" |
56949 | 115 |
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)" |
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using simple_function_borel_measurable[of f] measurable_simple_function[of M f] |
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by (fastforce simp: simple_function_def) |
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118 |
|
47694 | 119 |
lemma simple_function_const[intro, simp]: |
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"simple_function M (\<lambda>x. c)" |
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by (auto intro: finite_subset simp: simple_function_def) |
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lemma simple_function_compose[intro, simp]: |
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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;
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|
124 |
shows "simple_function M (g \<circ> f)" |
38656 | 125 |
unfolding simple_function_def |
126 |
proof safe |
|
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show "finite ((g \<circ> f) ` space M)" |
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using assms unfolding simple_function_def by (auto simp: image_comp [symmetric]) |
38656 | 129 |
next |
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fix x assume "x \<in> space M" |
|
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let ?G = "g -` {g (f x)} \<inter> (f`space M)" |
|
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have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M = |
|
133 |
(\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto |
|
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show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M" |
|
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using assms unfolding simple_function_def * |
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immler
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diff
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136 |
by (rule_tac sets.finite_UN) auto |
38656 | 137 |
qed |
138 |
||
47694 | 139 |
lemma simple_function_indicator[intro, simp]: |
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assumes "A \<in> sets M" |
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141 |
shows "simple_function M (indicator A)" |
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proof - |
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have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _") |
144 |
by (auto simp: indicator_def) |
|
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hence "finite ?S" by (rule finite_subset) simp |
|
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moreover have "- A \<inter> space M = space M - A" by auto |
|
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ultimately show ?thesis unfolding simple_function_def |
|
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using assms by (auto simp: indicator_def [abs_def]) |
35692 | 149 |
qed |
150 |
||
47694 | 151 |
lemma simple_function_Pair[intro, simp]: |
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152 |
assumes "simple_function M f" |
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diff
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153 |
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
|
154 |
shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p") |
38656 | 155 |
unfolding simple_function_def |
156 |
proof safe |
|
157 |
show "finite (?p ` space M)" |
|
158 |
using assms unfolding simple_function_def |
|
159 |
by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto |
|
160 |
next |
|
161 |
fix x assume "x \<in> space M" |
|
162 |
have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = |
|
163 |
(f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)" |
|
164 |
by auto |
|
165 |
with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M" |
|
166 |
using assms unfolding simple_function_def by auto |
|
167 |
qed |
|
35692 | 168 |
|
47694 | 169 |
lemma simple_function_compose1: |
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hoelzl
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170 |
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:
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diff
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|
171 |
shows "simple_function M (\<lambda>x. g (f x))" |
38656 | 172 |
using simple_function_compose[OF assms, of g] |
173 |
by (simp add: comp_def) |
|
35582 | 174 |
|
47694 | 175 |
lemma simple_function_compose2: |
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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176 |
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
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|
177 |
shows "simple_function M (\<lambda>x. h (f x) (g x))" |
38656 | 178 |
proof - |
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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diff
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|
179 |
have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))" |
38656 | 180 |
using assms by auto |
181 |
thus ?thesis by (simp_all add: comp_def) |
|
182 |
qed |
|
35582 | 183 |
|
47694 | 184 |
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"] |
38656 | 185 |
and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"] |
186 |
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"] |
|
187 |
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"] |
|
188 |
and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"] |
|
189 |
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"] |
|
41981
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190 |
and simple_function_max[intro, simp] = simple_function_compose2[where h=max] |
38656 | 191 |
|
47694 | 192 |
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
|
193 |
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
|
194 |
shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)" |
38656 | 195 |
proof cases |
196 |
assume "finite P" from this assms show ?thesis by induct auto |
|
197 |
qed auto |
|
35582 | 198 |
|
56949 | 199 |
lemma simple_function_ereal[intro, simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
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|
200 |
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f" |
56949 | 201 |
shows "simple_function M (\<lambda>x. ereal (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
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|
202 |
by (auto intro!: simple_function_compose1[OF sf]) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
203 |
|
56949 | 204 |
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
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|
205 |
fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f" |
56949 | 206 |
shows "simple_function M (\<lambda>x. real (f x))" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
207 |
by (auto intro!: simple_function_compose1[OF sf]) |
35582 | 208 |
|
47694 | 209 |
lemma borel_measurable_implies_simple_function_sequence: |
43920 | 210 |
fixes u :: "'a \<Rightarrow> ereal" |
38656 | 211 |
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
|
212 |
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
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|
213 |
(\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)" |
35582 | 214 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
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diff
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|
215 |
def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
216 |
{ 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
|
217 |
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
|
218 |
assume "\<not> real j \<le> u x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
219 |
then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)" |
56536 | 220 |
by (cases "u x") (auto intro!: natfloor_mono) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
221 |
moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j" |
56536 | 222 |
by (intro real_natfloor_le) auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
223 |
ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
224 |
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
|
225 |
qed auto } |
38656 | 226 |
note f_upper = this |
35582 | 227 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
228 |
have real_f: |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
229 |
"\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
230 |
unfolding f_def by auto |
35582 | 231 |
|
46731 | 232 |
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
|
233 |
show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
234 |
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
|
235 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
236 |
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
|
237 |
proof (intro simple_function_borel_measurable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
238 |
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
|
239 |
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
|
240 |
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
|
241 |
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
|
242 |
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
|
243 |
by (rule finite_subset) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
244 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
245 |
then show "simple_function M (?g i)" |
43920 | 246 |
by (auto intro: simple_function_ereal simple_function_div) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
247 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
248 |
show "incseq ?g" |
43920 | 249 |
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
|
250 |
fix x and i :: nat |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
251 |
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
|
252 |
proof ((split split_if)+, intro conjI impI) |
43920 | 253 |
assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
254 |
then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
255 |
by (cases "u x") (auto intro!: le_natfloor) |
38656 | 256 |
next |
43920 | 257 |
assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
258 |
then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
259 |
by (cases "u x") auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
260 |
next |
43920 | 261 |
assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
262 |
have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
263 |
by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
264 |
also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
265 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
266 |
assume "0 \<le> u x" then show ?thesis |
46671 | 267 |
by (intro le_mult_natfloor) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
268 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
269 |
assume "\<not> 0 \<le> u x" then show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
270 |
by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg) |
38656 | 271 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
272 |
also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
273 |
by (simp add: ac_simps) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
274 |
finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
275 |
qed simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
276 |
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
|
277 |
by (auto simp: field_simps) |
35582 | 278 |
qed |
38656 | 279 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
280 |
fix x show "(SUP i. ?g i x) = max 0 (u x)" |
51000 | 281 |
proof (rule SUP_eqI) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
282 |
fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
283 |
by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg |
56536 | 284 |
mult_nonpos_nonneg) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
285 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
286 |
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
|
287 |
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
|
288 |
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
|
289 |
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
|
290 |
proof (cases y) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
291 |
case (real r) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
292 |
with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq) |
44666 | 293 |
from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le) |
43920 | 294 |
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
|
295 |
then guess p .. note ux = this |
44666 | 296 |
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
|
297 |
have "p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
298 |
proof (rule ccontr) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
299 |
assume "\<not> p \<le> r" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
300 |
with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"] |
56536 | 301 |
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
|
302 |
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
|
303 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
304 |
have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
305 |
using *[of "max N m"] m unfolding real_f using ux |
56536 | 306 |
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
|
307 |
then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
308 |
by (metis real_natfloor_gt_diff_one less_le_trans) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
309 |
ultimately show False by auto |
38656 | 310 |
qed |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
311 |
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
|
312 |
qed (insert `0 \<le> y`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
313 |
qed |
56571
f4635657d66f
added divide_nonneg_nonneg and co; made it a simp rule
hoelzl
parents:
56537
diff
changeset
|
314 |
qed auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
315 |
qed |
35582 | 316 |
|
47694 | 317 |
lemma borel_measurable_implies_simple_function_sequence': |
43920 | 318 |
fixes u :: "'a \<Rightarrow> ereal" |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
319 |
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
|
320 |
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
|
321 |
"\<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
|
322 |
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
|
323 |
|
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
324 |
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
|
325 |
fixes u :: "'a \<Rightarrow> ereal" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
326 |
assumes u: "simple_function M u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
327 |
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
|
328 |
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
|
329 |
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
|
330 |
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
|
331 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
332 |
proof (rule cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
333 |
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
|
334 |
proof eventually_elim |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
335 |
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
|
336 |
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
|
337 |
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
|
338 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
339 |
next |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
340 |
from u have "finite (u ` space M)" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
341 |
unfolding simple_function_def by auto |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
342 |
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
|
343 |
proof induct |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
344 |
case empty show ?case |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
345 |
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
|
346 |
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
|
347 |
next |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
348 |
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
|
349 |
apply (subst simple_function_cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
350 |
apply (rule simple_function_indicator_representation[symmetric]) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
351 |
apply (auto intro: u) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
352 |
done |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
353 |
qed fact |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
354 |
|
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
355 |
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
|
356 |
fixes u :: "'a \<Rightarrow> ereal" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
357 |
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
|
358 |
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
|
359 |
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
|
360 |
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
|
361 |
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
|
362 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
363 |
proof - |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
364 |
show ?thesis |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
365 |
proof (rule cong) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
366 |
fix x assume x: "x \<in> space M" |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
367 |
from simple_function_indicator_representation[OF u x] |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
368 |
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
|
369 |
next |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
370 |
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
|
371 |
apply (subst simple_function_cong) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
372 |
apply (rule simple_function_indicator_representation[symmetric]) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
373 |
apply (auto intro: u) |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
374 |
done |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
375 |
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
|
376 |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
377 |
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
|
378 |
unfolding simple_function_def by auto |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
379 |
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
|
380 |
proof induct |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
381 |
case empty show ?case |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
382 |
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
|
383 |
next |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
384 |
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
|
385 |
{ 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
|
386 |
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
|
387 |
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
|
388 |
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
|
389 |
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
|
390 |
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
|
391 |
qed |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
392 |
qed fact |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
393 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
394 |
|
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
395 |
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
|
396 |
fixes u :: "'a \<Rightarrow> ereal" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
397 |
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
|
398 |
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
|
399 |
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
|
400 |
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
|
401 |
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
|
402 |
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
|
403 |
shows "P u" |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
404 |
using u |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
405 |
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
|
406 |
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
|
407 |
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
|
408 |
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
|
409 |
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
|
410 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
411 |
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
|
412 |
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
|
413 |
|
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
414 |
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
|
415 |
by (simp add: borel_measurable_simple_function) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
416 |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
417 |
show "P u" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
418 |
unfolding u_eq |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
419 |
proof (rule seq) |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
420 |
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
|
421 |
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
|
422 |
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
|
423 |
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
|
424 |
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
|
425 |
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
|
426 |
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
|
427 |
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
|
428 |
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
|
429 |
(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
|
430 |
next |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
431 |
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
|
432 |
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
|
433 |
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
|
434 |
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
|
435 |
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
|
436 |
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
|
437 |
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
|
438 |
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
|
439 |
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
|
440 |
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
|
441 |
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
|
442 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
443 |
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
|
444 |
qed fact+ |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
445 |
qed |
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
446 |
|
47694 | 447 |
lemma simple_function_If_set: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
448 |
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
|
449 |
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
|
450 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
451 |
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
|
452 |
show ?thesis unfolding simple_function_def |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
453 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
454 |
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
|
455 |
from finite_subset[OF this] assms |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
456 |
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
|
457 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
458 |
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
|
459 |
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
|
460 |
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
|
461 |
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
|
462 |
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
|
463 |
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
|
464 |
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
|
465 |
show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto |
35582 | 466 |
qed |
467 |
qed |
|
468 |
||
47694 | 469 |
lemma simple_function_If: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
470 |
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
|
471 |
shows "simple_function M (\<lambda>x. if P x then f x else g x)" |
35582 | 472 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
473 |
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
|
474 |
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp |
38656 | 475 |
qed |
476 |
||
47694 | 477 |
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
|
478 |
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
|
479 |
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
|
480 |
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
|
481 |
using assms unfolding simple_function_def by auto |
39092 | 482 |
|
47694 | 483 |
lemma simple_function_comp: |
484 |
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
|
485 |
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
|
486 |
shows "simple_function M (\<lambda>x. f (T x))" |
41661 | 487 |
proof (intro simple_function_def[THEN iffD2] conjI ballI) |
488 |
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'" |
|
489 |
using T unfolding measurable_def by auto |
|
490 |
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
|
491 |
using f unfolding simple_function_def by (auto intro: finite_subset) |
41661 | 492 |
fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M" |
493 |
then have "i \<in> f ` space M'" |
|
494 |
using T unfolding measurable_def by auto |
|
495 |
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
|
496 |
using f unfolding simple_function_def by auto |
41661 | 497 |
then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M" |
498 |
using T unfolding measurable_def by auto |
|
499 |
also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M" |
|
500 |
using T unfolding measurable_def by auto |
|
501 |
finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" . |
|
40859 | 502 |
qed |
503 |
||
56994 | 504 |
subsection "Simple integral" |
38656 | 505 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
506 |
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
|
507 |
"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
|
508 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
509 |
syntax |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
510 |
"_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
|
511 |
|
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
512 |
translations |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
513 |
"\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)" |
35582 | 514 |
|
47694 | 515 |
lemma simple_integral_cong: |
38656 | 516 |
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
|
517 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
38656 | 518 |
proof - |
519 |
have "f ` space M = g ` space M" |
|
520 |
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M" |
|
521 |
using assms by (auto intro!: image_eqI) |
|
522 |
thus ?thesis unfolding simple_integral_def by simp |
|
523 |
qed |
|
524 |
||
47694 | 525 |
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
|
526 |
"(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)" |
38656 | 527 |
proof (cases "space M = {}") |
528 |
case True thus ?thesis unfolding simple_integral_def by simp |
|
529 |
next |
|
530 |
case False hence "(\<lambda>x. c) ` space M = {c}" by auto |
|
531 |
thus ?thesis unfolding simple_integral_def by simp |
|
35582 | 532 |
qed |
533 |
||
47694 | 534 |
lemma simple_function_partition: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
535 |
assumes f: "simple_function M f" and g: "simple_function M g" |
56949 | 536 |
assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y" |
537 |
assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)" |
|
538 |
shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})" |
|
539 |
(is "_ = ?r") |
|
540 |
proof - |
|
541 |
from f g have [simp]: "finite (f`space M)" "finite (g`space M)" |
|
542 |
by (auto simp: simple_function_def) |
|
543 |
from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)" |
|
544 |
by (auto intro: measurable_simple_function) |
|
35582 | 545 |
|
56949 | 546 |
{ fix y assume "y \<in> space M" |
547 |
then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}" |
|
548 |
by (auto cong: sub simp: v[symmetric]) } |
|
549 |
note eq = this |
|
35582 | 550 |
|
56949 | 551 |
have "integral\<^sup>S M f = |
552 |
(\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M. |
|
553 |
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))" |
|
554 |
unfolding simple_integral_def |
|
555 |
proof (safe intro!: setsum_cong ereal_left_mult_cong) |
|
556 |
fix y assume y: "y \<in> space M" "f y \<noteq> 0" |
|
557 |
have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} = |
|
558 |
{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}" |
|
559 |
by auto |
|
560 |
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}) = |
|
561 |
f -` {f y} \<inter> space M" |
|
562 |
by (auto simp: eq_commute cong: sub rev_conj_cong) |
|
563 |
have "finite (g`space M)" by simp |
|
564 |
then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}" |
|
565 |
by (rule rev_finite_subset) auto |
|
566 |
then show "emeasure M (f -` {f y} \<inter> space M) = |
|
567 |
(\<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)" |
|
568 |
apply (simp add: setsum_cases) |
|
569 |
apply (subst setsum_emeasure) |
|
570 |
apply (auto simp: disjoint_family_on_def eq) |
|
571 |
done |
|
38656 | 572 |
qed |
56949 | 573 |
also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M. |
574 |
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))" |
|
575 |
by (auto intro!: setsum_cong simp: setsum_ereal_right_distrib emeasure_nonneg) |
|
576 |
also have "\<dots> = ?r" |
|
577 |
by (subst setsum_commute) |
|
578 |
(auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq) |
|
579 |
finally show "integral\<^sup>S M f = ?r" . |
|
35582 | 580 |
qed |
581 |
||
47694 | 582 |
lemma simple_integral_add[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
583 |
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
|
584 |
shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g" |
35582 | 585 |
proof - |
56949 | 586 |
have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = |
587 |
(\<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})" |
|
588 |
by (intro simple_function_partition) (auto intro: f g) |
|
589 |
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}) + |
|
590 |
(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})" |
|
591 |
using assms(2,4) by (auto intro!: setsum_cong ereal_left_distrib simp: setsum_addf[symmetric]) |
|
592 |
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)" |
|
593 |
by (intro simple_function_partition[symmetric]) (auto intro: f g) |
|
594 |
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)" |
|
595 |
by (intro simple_function_partition[symmetric]) (auto intro: f g) |
|
596 |
finally show ?thesis . |
|
35582 | 597 |
qed |
598 |
||
47694 | 599 |
lemma simple_integral_setsum[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
600 |
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
|
601 |
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
|
602 |
shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))" |
38656 | 603 |
proof cases |
604 |
assume "finite P" |
|
605 |
from this assms show ?thesis |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
606 |
by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg) |
38656 | 607 |
qed auto |
608 |
||
47694 | 609 |
lemma simple_integral_mult[simp]: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
610 |
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
|
611 |
shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f" |
38656 | 612 |
proof - |
56949 | 613 |
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})" |
614 |
using f by (intro simple_function_partition) auto |
|
615 |
also have "\<dots> = c * integral\<^sup>S M f" |
|
616 |
using f unfolding simple_integral_def |
|
617 |
by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult_assoc Int_def conj_commute) |
|
618 |
finally show ?thesis . |
|
40871 | 619 |
qed |
620 |
||
47694 | 621 |
lemma simple_integral_mono_AE: |
56949 | 622 |
assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g" |
47694 | 623 |
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
|
624 |
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g" |
40859 | 625 |
proof - |
56949 | 626 |
let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}" |
627 |
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))" |
|
628 |
using f g by (intro simple_function_partition) auto |
|
629 |
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))" |
|
630 |
proof (clarsimp intro!: setsum_mono) |
|
40859 | 631 |
fix x assume "x \<in> space M" |
56949 | 632 |
let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)" |
633 |
show "f x * ?M \<le> g x * ?M" |
|
634 |
proof cases |
|
635 |
assume "?M \<noteq> 0" |
|
636 |
then have "0 < ?M" |
|
637 |
by (simp add: less_le emeasure_nonneg) |
|
638 |
also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)" |
|
639 |
using mono by (intro emeasure_mono_AE) auto |
|
640 |
finally have "\<not> \<not> f x \<le> g x" |
|
641 |
by (intro notI) auto |
|
642 |
then show ?thesis |
|
643 |
by (intro ereal_mult_right_mono) auto |
|
644 |
qed simp |
|
40859 | 645 |
qed |
56949 | 646 |
also have "\<dots> = integral\<^sup>S M g" |
647 |
using f g by (intro simple_function_partition[symmetric]) auto |
|
648 |
finally show ?thesis . |
|
40859 | 649 |
qed |
650 |
||
47694 | 651 |
lemma simple_integral_mono: |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
652 |
assumes "simple_function M f" and "simple_function M g" |
38656 | 653 |
and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
654 |
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g" |
41705 | 655 |
using assms by (intro simple_integral_mono_AE) auto |
35582 | 656 |
|
47694 | 657 |
lemma simple_integral_cong_AE: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
658 |
assumes "simple_function M f" and "simple_function M g" |
47694 | 659 |
and "AE x in M. f x = g x" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
660 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
40859 | 661 |
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE) |
662 |
||
47694 | 663 |
lemma simple_integral_cong': |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
664 |
assumes sf: "simple_function M f" "simple_function M g" |
47694 | 665 |
and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
666 |
shows "integral\<^sup>S M f = integral\<^sup>S M g" |
40859 | 667 |
proof (intro simple_integral_cong_AE sf AE_I) |
47694 | 668 |
show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact |
40859 | 669 |
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M" |
670 |
using sf[THEN borel_measurable_simple_function] by auto |
|
671 |
qed simp |
|
672 |
||
47694 | 673 |
lemma simple_integral_indicator: |
56949 | 674 |
assumes A: "A \<in> sets M" |
49796
182fa22e7ee8
introduce induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49795
diff
changeset
|
675 |
assumes f: "simple_function M f" |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
676 |
shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = |
56949 | 677 |
(\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))" |
678 |
proof - |
|
679 |
have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A" |
|
680 |
using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm) |
|
681 |
have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}" |
|
682 |
by (auto simp: image_iff) |
|
683 |
||
684 |
have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = |
|
685 |
(\<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})" |
|
686 |
using assms by (intro simple_function_partition) auto |
|
687 |
also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M. |
|
688 |
if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)" |
|
689 |
by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum_cong) |
|
690 |
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))" |
|
691 |
using assms by (subst setsum_cases) (auto intro!: simple_functionD(1) simp: eq) |
|
692 |
also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))" |
|
693 |
by (subst setsum_reindex[where f=fst]) (auto simp: inj_on_def) |
|
694 |
also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))" |
|
695 |
using A[THEN sets.sets_into_space] |
|
696 |
by (intro setsum_mono_zero_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2) |
|
697 |
finally show ?thesis . |
|
38656 | 698 |
qed |
35582 | 699 |
|
47694 | 700 |
lemma simple_integral_indicator_only[simp]: |
38656 | 701 |
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
|
702 |
shows "integral\<^sup>S M (indicator A) = emeasure M A" |
56949 | 703 |
using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms] |
704 |
by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm) |
|
35582 | 705 |
|
47694 | 706 |
lemma simple_integral_null_set: |
707 |
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
|
708 |
shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0" |
38656 | 709 |
proof - |
47694 | 710 |
have "AE x in M. indicator N x = (0 :: ereal)" |
711 |
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
|
712 |
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
|
713 |
using assms apply (intro simple_integral_cong_AE) by auto |
40859 | 714 |
then show ?thesis by simp |
38656 | 715 |
qed |
35582 | 716 |
|
47694 | 717 |
lemma simple_integral_cong_AE_mult_indicator: |
718 |
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
|
719 |
shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)" |
41705 | 720 |
using assms by (intro simple_integral_cong_AE) auto |
35582 | 721 |
|
47694 | 722 |
lemma simple_integral_cmult_indicator: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
723 |
assumes A: "A \<in> sets M" |
56949 | 724 |
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
|
725 |
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
|
726 |
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
|
727 |
|
56996 | 728 |
lemma simple_integral_nonneg: |
47694 | 729 |
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
|
730 |
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
|
731 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
732 |
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
|
733 |
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
|
734 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
735 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
736 |
|
56994 | 737 |
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
|
738 |
|
56996 | 739 |
definition nn_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>N") where |
740 |
"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 | 741 |
|
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
742 |
syntax |
56996 | 743 |
"_nn_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110) |
41689
3e39b0e730d6
the measure valuation is again part of the measure_space type, instead of an explicit parameter to the locale;
hoelzl
parents:
41661
diff
changeset
|
744 |
|
3e39b0e730d6
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 |
translations |
56996 | 746 |
"\<integral>\<^sup>+x. f \<partial>M" == "CONST nn_integral M (\<lambda>x. f)" |
40872 | 747 |
|
56996 | 748 |
lemma nn_integral_nonneg: |
749 |
"0 \<le> integral\<^sup>N M f" |
|
750 |
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: nn_integral_def le_fun_def) |
|
40873 | 751 |
|
56996 | 752 |
lemma nn_integral_not_MInfty[simp]: "integral\<^sup>N M f \<noteq> -\<infinity>" |
753 |
using nn_integral_nonneg[of M f] by auto |
|
47694 | 754 |
|
56996 | 755 |
lemma nn_integral_def_finite: |
756 |
"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
|
757 |
(is "_ = SUPREMUM ?A ?f") |
56996 | 758 |
unfolding nn_integral_def |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
759 |
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
|
760 |
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
|
761 |
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
|
762 |
note gM = g(1)[THEN borel_measurable_simple_function] |
50252 | 763 |
have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto |
46731 | 764 |
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
|
765 |
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
|
766 |
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
|
767 |
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
|
768 |
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
|
769 |
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
|
770 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
771 |
have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto |
47694 | 772 |
assume "(emeasure M) ?G = 0" |
773 |
with gM have "AE x in M. x \<notin> ?G" |
|
774 |
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
|
775 |
with gM g show ?thesis |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
776 |
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
|
777 |
(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
|
778 |
next |
50252 | 779 |
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
|
780 |
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
|
781 |
proof (intro SUP_PInfty) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
782 |
fix n :: nat |
47694 | 783 |
let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)" |
50252 | 784 |
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
|
785 |
then have "?g ?y \<in> ?A" by (rule g_in_A) |
47694 | 786 |
have "real n \<le> ?y * (emeasure M) ?G" |
50252 | 787 |
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
|
788 |
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
|
789 |
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
|
790 |
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
|
791 |
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
|
792 |
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
|
793 |
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
|
794 |
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
|
795 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
796 |
then show ?thesis by simp |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
797 |
qed |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
798 |
qed (auto intro: SUP_upper) |
40873 | 799 |
|
56996 | 800 |
lemma nn_integral_mono_AE: |
801 |
assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v" |
|
802 |
unfolding nn_integral_def |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
803 |
proof (safe intro!: SUP_mono) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
804 |
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
|
805 |
from ae[THEN AE_E] guess N . note N = this |
47694 | 806 |
then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in) |
46731 | 807 |
let ?n = "\<lambda>x. n x * indicator (space M - N) x" |
47694 | 808 |
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
|
809 |
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
|
810 |
moreover |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
811 |
{ 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
|
812 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
813 |
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
|
814 |
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
|
815 |
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
|
816 |
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
|
817 |
qed simp } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
818 |
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
|
819 |
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
|
820 |
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
|
821 |
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
|
822 |
by force |
38656 | 823 |
qed |
824 |
||
56996 | 825 |
lemma nn_integral_mono: |
826 |
"(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>N M u \<le> integral\<^sup>N M v" |
|
827 |
by (auto intro: nn_integral_mono_AE) |
|
40859 | 828 |
|
56996 | 829 |
lemma nn_integral_cong_AE: |
830 |
"AE x in M. u x = v x \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v" |
|
831 |
by (auto simp: eq_iff intro!: nn_integral_mono_AE) |
|
40859 | 832 |
|
56996 | 833 |
lemma nn_integral_cong: |
834 |
"(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v" |
|
835 |
by (auto intro: nn_integral_cong_AE) |
|
40859 | 836 |
|
56996 | 837 |
lemma nn_integral_cong_strong: |
838 |
"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" |
|
839 |
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
|
840 |
|
56996 | 841 |
lemma nn_integral_eq_simple_integral: |
842 |
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
|
843 |
proof - |
46731 | 844 |
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
|
845 |
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
|
846 |
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
|
847 |
by (auto simp: fun_eq_iff max_def split: split_indicator) |
56996 | 848 |
have "integral\<^sup>N M ?f \<le> integral\<^sup>S M ?f" using f' |
849 |
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def nn_integral_def) |
|
850 |
moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>N M ?f" |
|
851 |
unfolding nn_integral_def |
|
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
852 |
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
|
853 |
ultimately show ?thesis |
56996 | 854 |
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
|
855 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
856 |
|
56996 | 857 |
lemma nn_integral_eq_simple_integral_AE: |
858 |
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
|
859 |
proof - |
47694 | 860 |
have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max) |
56996 | 861 |
with f have "integral\<^sup>N M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))" |
862 |
by (simp cong: nn_integral_cong_AE simple_integral_cong_AE |
|
863 |
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
|
864 |
with assms show ?thesis |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
865 |
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
|
866 |
qed |
40873 | 867 |
|
56996 | 868 |
lemma nn_integral_SUP_approx: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
869 |
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
|
870 |
and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}" |
56996 | 871 |
shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>N M (f i))" (is "_ \<le> ?S") |
43920 | 872 |
proof (rule ereal_le_mult_one_interval) |
56996 | 873 |
have "0 \<le> (SUP i. integral\<^sup>N M (f i))" |
874 |
using f(3) by (auto intro!: SUP_upper2 nn_integral_nonneg) |
|
875 |
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
|
876 |
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
|
877 |
using u(3) by auto |
43920 | 878 |
fix a :: ereal assume "0 < a" "a < 1" |
38656 | 879 |
hence "a \<noteq> 0" by auto |
46731 | 880 |
let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}" |
38656 | 881 |
have B: "\<And>i. ?B i \<in> sets M" |
56949 | 882 |
using f `simple_function M u`[THEN borel_measurable_simple_function] by auto |
38656 | 883 |
|
46731 | 884 |
let ?uB = "\<lambda>i x. u x * indicator (?B i) x" |
38656 | 885 |
|
886 |
{ fix i have "?B i \<subseteq> ?B (Suc i)" |
|
887 |
proof safe |
|
888 |
fix i x assume "a * u x \<le> f i x" |
|
889 |
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
|
890 |
using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto |
38656 | 891 |
finally show "a * u x \<le> f (Suc i) x" . |
892 |
qed } |
|
893 |
note B_mono = this |
|
35582 | 894 |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
895 |
note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B] |
38656 | 896 |
|
46731 | 897 |
let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n" |
47694 | 898 |
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
|
899 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
900 |
fix i |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
901 |
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
|
902 |
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
|
903 |
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
|
904 |
proof safe |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
905 |
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
|
906 |
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
|
907 |
proof cases |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
908 |
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
|
909 |
next |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
910 |
assume "u x \<noteq> 0" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
911 |
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
|
912 |
have "a * u x < 1 * u x" |
43920 | 913 |
by (intro ereal_mult_strict_right_mono) (auto simp: image_iff) |
46884 | 914 |
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
|
915 |
finally obtain i where "a * u x < f i x" unfolding SUP_def |
56166 | 916 |
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
|
917 |
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
|
918 |
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
|
919 |
qed |
40859 | 920 |
qed |
47694 | 921 |
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
|
922 |
qed |
38656 | 923 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
924 |
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
|
925 |
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
|
926 |
proof (subst SUP_ereal_setsum, safe) |
38656 | 927 |
fix x n assume "x \<in> space M" |
47694 | 928 |
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)" |
929 |
using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff) |
|
38656 | 930 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
931 |
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
|
932 |
using measure_conv u_range B_u unfolding simple_integral_def |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
933 |
by (auto intro!: setsum_cong SUP_ereal_cmult [symmetric]) |
38656 | 934 |
qed |
935 |
moreover |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
936 |
have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S" |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
937 |
apply (subst SUP_ereal_cmult [symmetric]) |
38705 | 938 |
proof (safe intro!: SUP_mono bexI) |
38656 | 939 |
fix i |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
940 |
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
|
941 |
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
|
942 |
by (subst simple_integral_mult) (auto split: split_indicator) |
56996 | 943 |
also have "\<dots> \<le> integral\<^sup>N M (f i)" |
38656 | 944 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
945 |
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
|
946 |
show ?thesis using f(3) * u_range `0 < a` |
56996 | 947 |
by (subst nn_integral_eq_simple_integral[symmetric]) |
948 |
(auto intro!: nn_integral_mono split: split_indicator) |
|
38656 | 949 |
qed |
56996 | 950 |
finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>N M (f i)" |
38656 | 951 |
by auto |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
952 |
next |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
953 |
fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range |
56996 | 954 |
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
|
955 |
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
|
956 |
ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp |
35582 | 957 |
qed |
958 |
||
56996 | 959 |
lemma incseq_nn_integral: |
960 |
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
|
961 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
962 |
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
|
963 |
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
|
964 |
then show ?thesis |
56996 | 965 |
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
|
966 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
967 |
|
35582 | 968 |
text {* Beppo-Levi monotone convergence theorem *} |
56996 | 969 |
lemma nn_integral_monotone_convergence_SUP: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
970 |
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x" |
56996 | 971 |
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
|
972 |
proof (rule antisym) |
56996 | 973 |
show "(SUP j. integral\<^sup>N M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)" |
974 |
by (auto intro!: SUP_least SUP_upper nn_integral_mono) |
|
38656 | 975 |
next |
56996 | 976 |
show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>N M (f j))" |
977 |
unfolding nn_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"] |
|
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
978 |
proof (safe intro!: SUP_least) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
979 |
fix g assume g: "simple_function M g" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
980 |
and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
981 |
then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}" |
44928
7ef6505bde7f
renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents:
44890
diff
changeset
|
982 |
using f by (auto intro!: SUP_upper2) |
56996 | 983 |
with * show "integral\<^sup>S M g \<le> (SUP j. integral\<^sup>N M (f j))" |
984 |
by (intro nn_integral_SUP_approx[OF f g _ g']) |
|
46884 | 985 |
(auto simp: le_fun_def max_def) |
35582 | 986 |
qed |
987 |
qed |
|
988 |
||
56996 | 989 |
lemma nn_integral_monotone_convergence_SUP_AE: |
47694 | 990 |
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 | 991 |
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))" |
40859 | 992 |
proof - |
47694 | 993 |
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
|
994 |
by (simp add: AE_all_countable) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
995 |
from this[THEN AE_E] guess N . note N = this |
46731 | 996 |
let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0" |
47694 | 997 |
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
|
998 |
then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)" |
56996 | 999 |
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
|
1000 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))" |
56996 | 1001 |
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
|
1002 |
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
|
1003 |
{ fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1004 |
using f N(3) by (intro measurable_If_set) auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1005 |
fix x show "0 \<le> ?f i x" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1006 |
using N(1) by auto } |
40859 | 1007 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1008 |
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))" |
56996 | 1009 |
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
|
1010 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1011 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1012 |
|
56996 | 1013 |
lemma nn_integral_monotone_convergence_SUP_AE_incseq: |
47694 | 1014 |
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 | 1015 |
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
|
1016 |
using f[unfolded incseq_Suc_iff le_fun_def] |
56996 | 1017 |
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
|
1018 |
auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1019 |
|
56996 | 1020 |
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
|
1021 |
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
|
1022 |
shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)" |
56996 | 1023 |
using assms unfolding nn_integral_monotone_convergence_SUP[OF f(1) |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1024 |
f(3)[THEN borel_measurable_simple_function] f(2)] |
56996 | 1025 |
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
|
1026 |
|
56996 | 1027 |
lemma nn_integral_max_0: |
1028 |
"(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>N M f" |
|
1029 |
by (simp add: le_fun_def nn_integral_def) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1030 |
|
56996 | 1031 |
lemma nn_integral_cong_pos: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1032 |
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x" |
56996 | 1033 |
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
|
1034 |
proof - |
56996 | 1035 |
have "integral\<^sup>N M (\<lambda>x. max 0 (f x)) = integral\<^sup>N M (\<lambda>x. max 0 (g x))" |
1036 |
proof (intro nn_integral_cong) |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1037 |
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
|
1038 |
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
|
1039 |
by (auto split: split_max) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1040 |
qed |
56996 | 1041 |
then show ?thesis by (simp add: nn_integral_max_0) |
40859 | 1042 |
qed |
1043 |
||
47694 | 1044 |
lemma SUP_simple_integral_sequences: |
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)" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1046 |
and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)" |
47694 | 1047 |
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
|
1048 |
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
|
1049 |
(is "SUPREMUM _ ?F = SUPREMUM _ ?G") |
38656 | 1050 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1051 |
have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)" |
56996 | 1052 |
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
|
1053 |
also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)" |
56996 | 1054 |
unfolding eq[THEN nn_integral_cong_AE] .. |
38656 | 1055 |
also have "\<dots> = (SUP i. ?G i)" |
56996 | 1056 |
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
|
1057 |
finally show ?thesis by simp |
38656 | 1058 |
qed |
1059 |
||
56996 | 1060 |
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
|
1061 |
"0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)" |
56996 | 1062 |
by (subst nn_integral_eq_simple_integral) auto |
38656 | 1063 |
|
56996 | 1064 |
lemma nn_integral_linear: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1065 |
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
|
1066 |
and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x" |
56996 | 1067 |
shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>N M f + integral\<^sup>N M g" |
1068 |
(is "integral\<^sup>N M ?L = _") |
|
35582 | 1069 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1070 |
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u . |
56996 | 1071 |
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
|
1072 |
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v . |
56996 | 1073 |
note v = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this |
46731 | 1074 |
let ?L' = "\<lambda>i x. a * u i x + v i x" |
38656 | 1075 |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1076 |
have "?L \<in> borel_measurable M" using assms by auto |
38656 | 1077 |
from borel_measurable_implies_simple_function_sequence'[OF this] guess l . |
56996 | 1078 |
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
|
1079 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1080 |
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
|
1081 |
using u v `0 \<le> a` |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1082 |
by (auto simp: incseq_Suc_iff le_fun_def |
43920 | 1083 |
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
|
1084 |
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 | 1085 |
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
|
1086 |
{ 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
|
1087 |
by (auto split: split_if_asm) } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1088 |
note not_MInf = this |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1089 |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1090 |
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
|
1091 |
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
|
1092 |
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
|
1093 |
using u v `0 \<le> a` unfolding incseq_Suc_iff le_fun_def |
56537 | 1094 |
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
|
1095 |
{ fix x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1096 |
{ 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
|
1097 |
by auto } |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1098 |
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
|
1099 |
using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x] |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1100 |
by (subst SUP_ereal_cmult [symmetric, OF u(6) `0 \<le> a`]) |
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1101 |
(auto intro!: SUP_ereal_add |
56537 | 1102 |
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) } |
47694 | 1103 |
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
|
1104 |
unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2) |
56537 | 1105 |
by (intro AE_I2) (auto split: split_max) |
38656 | 1106 |
qed |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1107 |
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
|
1108 |
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
|
1109 |
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
|
1110 |
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
|
1111 |
unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric] |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1112 |
apply (subst SUP_ereal_cmult [symmetric, OF pos(1) `0 \<le> a`]) |
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1113 |
apply (subst SUP_ereal_add [symmetric, OF inc not_MInf]) . |
56996 | 1114 |
then show ?thesis by (simp add: nn_integral_max_0) |
38656 | 1115 |
qed |
1116 |
||
56996 | 1117 |
lemma nn_integral_cmult: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1118 |
assumes f: "f \<in> borel_measurable M" "0 \<le> c" |
56996 | 1119 |
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
|
1120 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1121 |
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c` |
43920 | 1122 |
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
|
1123 |
have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)" |
56996 | 1124 |
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
|
1125 |
then show ?thesis |
56996 | 1126 |
using nn_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f |
1127 |
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
|
1128 |
qed |
38656 | 1129 |
|
56996 | 1130 |
lemma nn_integral_multc: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1131 |
assumes "f \<in> borel_measurable M" "0 \<le> c" |
56996 | 1132 |
shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>N M f * c" |
1133 |
unfolding mult_commute[of _ c] nn_integral_cmult[OF assms] by simp |
|
41096 | 1134 |
|
56996 | 1135 |
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
|
1136 |
"A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A" |
56996 | 1137 |
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
|
1138 |
(auto simp: simple_integral_indicator) |
38656 | 1139 |
|
56996 | 1140 |
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
|
1141 |
"0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A" |
56996 | 1142 |
by (subst nn_integral_eq_simple_integral) |
41544 | 1143 |
(auto simp: simple_function_indicator simple_integral_indicator) |
38656 | 1144 |
|
56996 | 1145 |
lemma nn_integral_indicator': |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1146 |
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
|
1147 |
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
|
1148 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1149 |
have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)" |
56996 | 1150 |
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
|
1151 |
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
|
1152 |
by simp |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1153 |
finally show ?thesis . |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1154 |
qed |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1155 |
|
56996 | 1156 |
lemma nn_integral_add: |
47694 | 1157 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
1158 |
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
56996 | 1159 |
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
|
1160 |
proof - |
47694 | 1161 |
have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)" |
56537 | 1162 |
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
|
1163 |
have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)" |
56996 | 1164 |
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
|
1165 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)" |
56996 | 1166 |
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
|
1167 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)" |
56996 | 1168 |
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
|
1169 |
by auto |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1170 |
finally show ?thesis |
56996 | 1171 |
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
|
1172 |
qed |
38656 | 1173 |
|
56996 | 1174 |
lemma nn_integral_setsum: |
47694 | 1175 |
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 | 1176 |
shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>N M (f i))" |
38656 | 1177 |
proof cases |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1178 |
assume f: "finite P" |
47694 | 1179 |
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
|
1180 |
from f this assms(1) show ?thesis |
38656 | 1181 |
proof induct |
1182 |
case (insert i P) |
|
47694 | 1183 |
then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x" |
1184 |
"(\<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
|
1185 |
by (auto intro!: setsum_nonneg) |
56996 | 1186 |
from nn_integral_add[OF this] |
38656 | 1187 |
show ?case using insert by auto |
1188 |
qed simp |
|
1189 |
qed simp |
|
1190 |
||
56996 | 1191 |
lemma nn_integral_Markov_inequality: |
49775
970964690b3d
remove some unneeded positivity assumptions; generalize some assumptions to AE; tuned proofs
hoelzl
parents:
47761
diff
changeset
|
1192 |
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
|
1193 |
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 | 1194 |
(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
|
1195 |
proof - |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1196 |
have "?A \<in> sets M" |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1197 |
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
|
1198 |
hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)" |
56996 | 1199 |
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
|
1200 |
also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c |
56996 | 1201 |
by (auto intro!: nn_integral_mono_AE |
43920 | 1202 |
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
|
1203 |
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
|
1204 |
using assms |
56996 | 1205 |
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
|
1206 |
finally show ?thesis . |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1207 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1208 |
|
56996 | 1209 |
lemma nn_integral_noteq_infinite: |
47694 | 1210 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
56996 | 1211 |
and "integral\<^sup>N M g \<noteq> \<infinity>" |
47694 | 1212 |
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
|
1213 |
proof (rule ccontr) |
47694 | 1214 |
assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)" |
1215 |
have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0" |
|
1216 |
using c g by (auto simp add: AE_iff_null) |
|
1217 |
moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets) |
|
1218 |
ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto |
|
1219 |
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
|
1220 |
also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)" |
56996 | 1221 |
using g by (subst nn_integral_cmult_indicator) auto |
1222 |
also have "\<dots> \<le> integral\<^sup>N M g" |
|
1223 |
using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def) |
|
1224 |
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
|
1225 |
qed |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1226 |
|
56996 | 1227 |
lemma nn_integral_PInf: |
56949 | 1228 |
assumes f: "f \<in> borel_measurable M" |
56996 | 1229 |
and not_Inf: "integral\<^sup>N M f \<noteq> \<infinity>" |
56949 | 1230 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
1231 |
proof - |
|
1232 |
have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)" |
|
56996 | 1233 |
using f by (subst nn_integral_cmult_indicator) (auto simp: measurable_sets) |
1234 |
also have "\<dots> \<le> integral\<^sup>N M (\<lambda>x. max 0 (f x))" |
|
1235 |
by (auto intro!: nn_integral_mono simp: indicator_def max_def) |
|
1236 |
finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>N M f" |
|
1237 |
by (simp add: nn_integral_max_0) |
|
56949 | 1238 |
moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)" |
1239 |
by (rule emeasure_nonneg) |
|
1240 |
ultimately show ?thesis |
|
1241 |
using assms by (auto split: split_if_asm) |
|
1242 |
qed |
|
1243 |
||
56996 | 1244 |
lemma nn_integral_PInf_AE: |
1245 |
assumes "f \<in> borel_measurable M" "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>" |
|
56949 | 1246 |
proof (rule AE_I) |
1247 |
show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
|
56996 | 1248 |
by (rule nn_integral_PInf[OF assms]) |
56949 | 1249 |
show "f -` {\<infinity>} \<inter> space M \<in> sets M" |
1250 |
using assms by (auto intro: borel_measurable_vimage) |
|
1251 |
qed auto |
|
1252 |
||
1253 |
lemma simple_integral_PInf: |
|
1254 |
assumes "simple_function M f" "\<And>x. 0 \<le> f x" |
|
1255 |
and "integral\<^sup>S M f \<noteq> \<infinity>" |
|
1256 |
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0" |
|
56996 | 1257 |
proof (rule nn_integral_PInf) |
56949 | 1258 |
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function) |
56996 | 1259 |
show "integral\<^sup>N M f \<noteq> \<infinity>" |
1260 |
using assms by (simp add: nn_integral_eq_simple_integral) |
|
56949 | 1261 |
qed |
1262 |
||
56996 | 1263 |
lemma nn_integral_diff: |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1264 |
assumes f: "f \<in> borel_measurable M" |
47694 | 1265 |
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
56996 | 1266 |
and fin: "integral\<^sup>N M g \<noteq> \<infinity>" |
47694 | 1267 |
and mono: "AE x in M. g x \<le> f x" |
56996 | 1268 |
shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>N M f - integral\<^sup>N M g" |
38656 | 1269 |
proof - |
47694 | 1270 |
have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x" |
43920 | 1271 |
using assms by (auto intro: ereal_diff_positive) |
47694 | 1272 |
have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto |
43920 | 1273 |
{ fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b" |
1274 |
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
|
1275 |
note * = this |
47694 | 1276 |
then have "AE x in M. f x = f x - g x + g x" |
56996 | 1277 |
using mono nn_integral_noteq_infinite[OF g fin] assms by auto |
1278 |
then have **: "integral\<^sup>N M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>N M g" |
|
1279 |
unfolding nn_integral_add[OF diff g, symmetric] |
|
1280 |
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
|
1281 |
show ?thesis unfolding ** |
56996 | 1282 |
using fin nn_integral_nonneg[of M g] |
1283 |
by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>N M g"]) auto |
|
38656 | 1284 |
qed |
1285 |
||
56996 | 1286 |
lemma nn_integral_suminf: |
47694 | 1287 |
assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x" |
56996 | 1288 |
shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>N M (f i))" |
38656 | 1289 |
proof - |
47694 | 1290 |
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
|
1291 |
using assms by (auto simp: AE_all_countable) |
56996 | 1292 |
have "(\<Sum>i. integral\<^sup>N M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>N M (f i))" |
1293 |
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
|
1294 |
also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)" |
56996 | 1295 |
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
|
1296 |
also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos |
56996 | 1297 |
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
|
1298 |
(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
|
1299 |
also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos |
56996 | 1300 |
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
|
1301 |
finally show ?thesis by simp |
38656 | 1302 |
qed |
1303 |
||
56996 | 1304 |
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
|
1305 |
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
|
1306 |
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
|
1307 |
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
|
1308 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1309 |
have "(\<integral>\<^sup>+x. g x \<partial>M) \<le> (\<integral>\<^sup>+x. c * f x \<partial>M)" |
56996 | 1310 |
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
|
1311 |
also have "(\<integral>\<^sup>+x. c * f x \<partial>M) < \<infinity>" |
56996 | 1312 |
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
|
1313 |
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
|
1314 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1315 |
|
38656 | 1316 |
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
|
1317 |
|
56996 | 1318 |
lemma nn_integral_liminf: |
43920 | 1319 |
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal" |
47694 | 1320 |
assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x" |
56996 | 1321 |
shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))" |
38656 | 1322 |
proof - |
47694 | 1323 |
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
|
1324 |
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
|
1325 |
(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
|
1326 |
unfolding liminf_SUP_INF using pos u |
56996 | 1327 |
by (intro nn_integral_monotone_convergence_SUP_AE) |
44937
22c0857b8aab
removed further legacy rules from Complete_Lattices
hoelzl
parents:
44928
diff
changeset
|
1328 |
(elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono) |
56996 | 1329 |
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
|
1330 |
unfolding liminf_SUP_INF |
56996 | 1331 |
by (auto intro!: SUP_mono exI INF_greatest nn_integral_mono INF_lower) |
38656 | 1332 |
finally show ?thesis . |
35582 | 1333 |
qed |
1334 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1335 |
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
|
1336 |
"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
|
1337 |
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
|
1338 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1339 |
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
|
1340 |
"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
|
1341 |
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
|
1342 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1343 |
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
|
1344 |
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
|
1345 |
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
|
1346 |
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
|
1347 |
|
56996 | 1348 |
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
|
1349 |
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
|
1350 |
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
|
1351 |
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 | 1352 |
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
|
1353 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1354 |
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
|
1355 |
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
|
1356 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1357 |
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
|
1358 |
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
|
1359 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1360 |
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 | 1361 |
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
|
1362 |
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
|
1363 |
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
|
1364 |
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
|
1365 |
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
|
1366 |
then have "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) < \<infinity>" |
56996 | 1367 |
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
|
1368 |
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
|
1369 |
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
|
1370 |
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
|
1371 |
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
|
1372 |
using w_nonneg |
56996 | 1373 |
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
|
1374 |
(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
|
1375 |
also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M)" |
56996 | 1376 |
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
|
1377 |
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
|
1378 |
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
|
1379 |
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
|
1380 |
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 | 1381 |
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
|
1382 |
fix i have "(\<integral>\<^sup>+x. u i x \<partial>M) < \<infinity>" |
56996 | 1383 |
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
|
1384 |
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
|
1385 |
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
|
1386 |
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
|
1387 |
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
|
1388 |
finally show ?thesis |
56996 | 1389 |
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
|
1390 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1391 |
|
56996 | 1392 |
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
|
1393 |
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
|
1394 |
"\<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
|
1395 |
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
|
1396 |
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
|
1397 |
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
|
1398 |
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
|
1399 |
proof - |
56996 | 1400 |
have "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)" |
1401 |
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
|
1402 |
moreover have "(\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)" |
56996 | 1403 |
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
|
1404 |
moreover have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)" |
56996 | 1405 |
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot) |
1406 |
moreover have "(\<integral>\<^sup>+x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))" |
|
1407 |
by (intro nn_integral_liminf[OF _ bound(1)]) auto |
|
1408 |
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
|
1409 |
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
|
1410 |
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
|
1411 |
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
|
1412 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56949
diff
changeset
|
1413 |
|
56996 | 1414 |
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
|
1415 |
assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0" |
38656 | 1416 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1417 |
have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)" |
56996 | 1418 |
proof (intro nn_integral_cong_AE AE_I) |
40859 | 1419 |
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N" |
1420 |
by (auto simp: indicator_def) |
|
47694 | 1421 |
show "(emeasure M) N = 0" "N \<in> sets M" |
40859 | 1422 |
using assms by auto |
35582 | 1423 |
qed |
40859 | 1424 |
then show ?thesis by simp |
38656 | 1425 |
qed |
35582 | 1426 |
|
56996 | 1427 |
lemma nn_integral_0_iff: |
47694 | 1428 |
assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x" |
56996 | 1429 |
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0" |
47694 | 1430 |
(is "_ \<longleftrightarrow> (emeasure M) ?A = 0") |
35582 | 1431 |
proof - |
56996 | 1432 |
have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>N M u" |
1433 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
|
38656 | 1434 |
show ?thesis |
1435 |
proof |
|
47694 | 1436 |
assume "(emeasure M) ?A = 0" |
56996 | 1437 |
with nn_integral_null_set[of ?A M u] u |
1438 |
show "integral\<^sup>N M u = 0" by (simp add: u_eq null_sets_def) |
|
38656 | 1439 |
next |
43920 | 1440 |
{ fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r" |
1441 |
then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def) |
|
1442 |
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
|
1443 |
note gt_1 = this |
56996 | 1444 |
assume *: "integral\<^sup>N M u = 0" |
46731 | 1445 |
let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}" |
47694 | 1446 |
have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))" |
38656 | 1447 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1448 |
{ fix n :: nat |
56996 | 1449 |
from nn_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"] |
47694 | 1450 |
have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp |
1451 |
moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto |
|
1452 |
ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto } |
|
38656 | 1453 |
thus ?thesis by simp |
35582 | 1454 |
qed |
47694 | 1455 |
also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)" |
1456 |
proof (safe intro!: SUP_emeasure_incseq) |
|
38656 | 1457 |
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
|
1458 |
using u by (auto intro!: sets.Int) |
38656 | 1459 |
next |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1460 |
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
|
1461 |
proof (safe intro!: incseq_SucI) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1462 |
fix n :: nat and x |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1463 |
assume *: "1 \<le> real n * u x" |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1464 |
also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x" |
43920 | 1465 |
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
|
1466 |
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
|
1467 |
qed |
38656 | 1468 |
qed |
47694 | 1469 |
also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}" |
1470 |
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
|
1471 |
fix x assume "0 < u x" and [simp, intro]: "x \<in> space M" |
38656 | 1472 |
show "x \<in> (\<Union>n. ?M n \<inter> ?A)" |
1473 |
proof (cases "u x") |
|
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1474 |
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
|
1475 |
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
|
1476 |
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
|
1477 |
hence "1 \<le> real j * r" using real `0 < r` by auto |
43920 | 1478 |
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
|
1479 |
qed (insert `0 < u x`, auto) |
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1480 |
qed auto |
47694 | 1481 |
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
|
1482 |
moreover |
47694 | 1483 |
from pos have "AE x in M. \<not> (u x < 0)" by auto |
1484 |
then have "(emeasure M) {x\<in>space M. u x < 0} = 0" |
|
1485 |
using AE_iff_null[of M] u by auto |
|
1486 |
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}" |
|
1487 |
using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"]) |
|
1488 |
ultimately show "(emeasure M) ?A = 0" by simp |
|
35582 | 1489 |
qed |
1490 |
qed |
|
1491 |
||
56996 | 1492 |
lemma nn_integral_0_iff_AE: |
41705 | 1493 |
assumes u: "u \<in> borel_measurable M" |
56996 | 1494 |
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)" |
41705 | 1495 |
proof - |
41981
cdf7693bbe08
reworked Probability theory: measures are not type restricted to positive extended reals
hoelzl
parents:
41831
diff
changeset
|
1496 |
have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M" |
41705 | 1497 |
using u by auto |
56996 | 1498 |
from nn_integral_0_iff[of "\<lambda>x. max 0 (u x)"] |
1499 |
have "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)" |
|
1500 |
unfolding nn_integral_max_0 |
|
47694 | 1501 |
using AE_iff_null[OF sets] u by auto |
1502 |
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
|
1503 |
finally show ?thesis . |
41705 | 1504 |
qed |
1505 |
||
56996 | 1506 |
lemma AE_iff_nn_integral: |
1507 |
"{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" |
|
1508 |
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
|
1509 |
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
|
1510 |
|
56996 | 1511 |
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
|
1512 |
"(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)" |
56996 | 1513 |
by (auto intro!: nn_integral_0_iff_AE[THEN iffD2]) |
42991
3fa22920bf86
integral strong monotone; finite subadditivity for measure
hoelzl
parents:
42950
diff
changeset
|
1514 |
|
56996 | 1515 |
lemma nn_integral_subalgebra: |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1516 |
assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x" |
47694 | 1517 |
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 | 1518 |
shows "integral\<^sup>N N f = integral\<^sup>N M f" |
39092 | 1519 |
proof - |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1520 |
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
|
1521 |
using N by (auto simp: measurable_def) |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1522 |
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
|
1523 |
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
|
1524 |
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
|
1525 |
using N by auto |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1526 |
from f show ?thesis |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1527 |
apply induct |
56996 | 1528 |
apply (simp_all add: nn_integral_add nn_integral_cmult nn_integral_monotone_convergence_SUP N) |
1529 |
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
|
1530 |
done |
39092 | 1531 |
qed |
1532 |
||
56996 | 1533 |
lemma nn_integral_nat_function: |
50097
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1534 |
fixes f :: "'a \<Rightarrow> nat" |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1535 |
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
|
1536 |
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
|
1537 |
proof - |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1538 |
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
|
1539 |
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
|
1540 |
by auto |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1541 |
|
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1542 |
{ 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
|
1543 |
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
|
1544 |
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
|
1545 |
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
|
1546 |
unfolding sums_ereal . |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1547 |
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
|
1548 |
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
|
1549 |
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
|
1550 |
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
|
1551 |
then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)" |
56996 | 1552 |
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
|
1553 |
also have "\<dots> = (\<Sum>i. emeasure M (F i))" |
56996 | 1554 |
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
|
1555 |
finally show ?thesis |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1556 |
by (simp add: F_def) |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1557 |
qed |
32973da2d4f7
rules for intergration: integrating nat-functions, integrals on finite measures, constant multiplication
hoelzl
parents:
50027
diff
changeset
|
1558 |
|
56994 | 1559 |
subsection {* Integral under concrete measures *} |
1560 |
||
1561 |
subsubsection {* Distributions *} |
|
47694 | 1562 |
|
56996 | 1563 |
lemma nn_integral_distr': |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1564 |
assumes T: "T \<in> measurable M M'" |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1565 |
and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x" |
56996 | 1566 |
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
|
1567 |
using f |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1568 |
proof induct |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1569 |
case (cong f g) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1570 |
with T show ?case |
56996 | 1571 |
apply (subst nn_integral_cong[of _ f g]) |
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1572 |
apply simp |
56996 | 1573 |
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
|
1574 |
apply (simp add: measurable_def Pi_iff) |
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1575 |
apply simp |
49797
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1576 |
done |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1577 |
next |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1578 |
case (set A) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1579 |
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
|
1580 |
by (auto simp: indicator_def) |
28066863284c
add induction rules for simple functions and for Borel measurable functions
hoelzl
parents:
49796
diff
changeset
|
1581 |
from set T show ?case |
56996 | 1582 |
by (subst nn_integral_cong[OF eq]) |
1583 |
(auto simp add: emeasure_distr intro!: nn_integral_indicator[symmetric] measurable_sets) |
|
1584 |
qed (simp_all add: measurable_compose[OF T] T nn_integral_cmult nn_integral_add |
|
1585 |
nn_integral_monotone_convergence_SUP le_fun_def incseq_def) |
|
47694 | 1586 |
|
56996 | 1587 |
lemma nn_integral_distr: |
1588 |
"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)" |
|
1589 |
by (subst (1 2) nn_integral_max_0[symmetric]) |
|
1590 |
(simp add: nn_integral_distr') |
|
35692 | 1591 |
|
56994 | 1592 |
subsubsection {* Counting space *} |
47694 | 1593 |
|
1594 |
lemma simple_function_count_space[simp]: |
|
1595 |
"simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)" |
|
1596 |
unfolding simple_function_def by simp |
|
1597 |
||
56996 | 1598 |
lemma nn_integral_count_space: |
47694 | 1599 |
assumes A: "finite {a\<in>A. 0 < f a}" |
56996 | 1600 |
shows "integral\<^sup>N (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
35582 | 1601 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1602 |
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
|
1603 |
(\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)" |
56996 | 1604 |
by (auto intro!: nn_integral_cong |
47694 | 1605 |
simp add: indicator_def if_distrib setsum_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
|
1606 |
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 | 1607 |
by (subst nn_integral_setsum) |
47694 | 1608 |
(simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le) |
1609 |
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)" |
|
56996 | 1610 |
by (auto intro!: setsum_cong simp: nn_integral_cmult_indicator one_ereal_def[symmetric]) |
1611 |
finally show ?thesis by (simp add: nn_integral_max_0) |
|
47694 | 1612 |
qed |
1613 |
||
56996 | 1614 |
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
|
1615 |
"finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))" |
56996 | 1616 |
by (subst nn_integral_max_0[symmetric]) |
1617 |
(auto intro!: setsum_mono_zero_left simp: nn_integral_count_space less_le) |
|
47694 | 1618 |
|
54418 | 1619 |
lemma emeasure_UN_countable: |
1620 |
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I: "countable I" |
|
1621 |
assumes disj: "disjoint_family_on X I" |
|
1622 |
shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)" |
|
1623 |
proof cases |
|
1624 |
assume "finite I" with sets disj show ?thesis |
|
1625 |
by (subst setsum_emeasure[symmetric]) |
|
56996 | 1626 |
(auto intro!: setsum_cong simp add: max_def subset_eq nn_integral_count_space_finite emeasure_nonneg) |
54418 | 1627 |
next |
1628 |
assume f: "\<not> finite I" |
|
1629 |
then have [intro]: "I \<noteq> {}" by auto |
|
1630 |
from from_nat_into_inj_infinite[OF I f] from_nat_into[OF this] disj |
|
1631 |
have disj2: "disjoint_family (\<lambda>i. X (from_nat_into I i))" |
|
1632 |
unfolding disjoint_family_on_def by metis |
|
1633 |
||
1634 |
from f have "bij_betw (from_nat_into I) UNIV I" |
|
1635 |
using bij_betw_from_nat_into[OF I] by simp |
|
1636 |
then have "(\<Union>i\<in>I. X i) = (\<Union>i. (X \<circ> from_nat_into I) i)" |
|
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
54611
diff
changeset
|
1637 |
unfolding SUP_def image_comp [symmetric] by (simp add: bij_betw_def) |
54418 | 1638 |
then have "emeasure M (UNION I X) = emeasure M (\<Union>i. X (from_nat_into I i))" |
1639 |
by simp |
|
1640 |
also have "\<dots> = (\<Sum>i. emeasure M (X (from_nat_into I i)))" |
|
1641 |
by (intro suminf_emeasure[symmetric] disj disj2) (auto intro!: sets from_nat_into[OF `I \<noteq> {}`]) |
|
1642 |
also have "\<dots> = (\<Sum>n. \<integral>\<^sup>+i. emeasure M (X i) * indicator {from_nat_into I n} i \<partial>count_space I)" |
|
1643 |
proof (intro arg_cong[where f=suminf] ext) |
|
1644 |
fix i |
|
1645 |
have eq: "{a \<in> I. 0 < emeasure M (X a) * indicator {from_nat_into I i} a} |
|
1646 |
= (if 0 < emeasure M (X (from_nat_into I i)) then {from_nat_into I i} else {})" |
|
1647 |
using ereal_0_less_1 |
|
1648 |
by (auto simp: ereal_zero_less_0_iff indicator_def from_nat_into `I \<noteq> {}` simp del: ereal_0_less_1) |
|
1649 |
have "(\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I) = |
|
1650 |
(if 0 < emeasure M (X (from_nat_into I i)) then emeasure M (X (from_nat_into I i)) else 0)" |
|
56996 | 1651 |
by (subst nn_integral_count_space) (simp_all add: eq) |
54418 | 1652 |
also have "\<dots> = emeasure M (X (from_nat_into I i))" |
1653 |
by (simp add: less_le emeasure_nonneg) |
|
1654 |
finally show "emeasure M (X (from_nat_into I i)) = |
|
1655 |
\<integral>\<^sup>+ ia. emeasure M (X ia) * indicator {from_nat_into I i} ia \<partial>count_space I" .. |
|
1656 |
qed |
|
1657 |
also have "\<dots> = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)" |
|
56996 | 1658 |
apply (subst nn_integral_suminf[symmetric]) |
1659 |
apply (auto simp: emeasure_nonneg intro!: nn_integral_cong) |
|
54418 | 1660 |
proof - |
1661 |
fix x assume "x \<in> I" |
|
1662 |
then have "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = (\<Sum>i\<in>{to_nat_on I x}. emeasure M (X x) * indicator {from_nat_into I i} x)" |
|
1663 |
by (intro suminf_finite) (auto simp: indicator_def I f) |
|
1664 |
also have "\<dots> = emeasure M (X x)" |
|
1665 |
by (simp add: I f `x\<in>I`) |
|
1666 |
finally show "(\<Sum>i. emeasure M (X x) * indicator {from_nat_into I i} x) = emeasure M (X x)" . |
|
1667 |
qed |
|
1668 |
finally show ?thesis . |
|
1669 |
qed |
|
1670 |
||
56994 | 1671 |
subsubsection {* Measures with Restricted Space *} |
54417 | 1672 |
|
56996 | 1673 |
lemma nn_integral_restrict_space: |
54417 | 1674 |
assumes \<Omega>: "\<Omega> \<in> sets M" and f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "\<And>x. x \<in> space M - \<Omega> \<Longrightarrow> f x = 0" |
56996 | 1675 |
shows "nn_integral (restrict_space M \<Omega>) f = nn_integral M f" |
54417 | 1676 |
using f proof (induct rule: borel_measurable_induct) |
1677 |
case (cong f g) then show ?case |
|
56996 | 1678 |
using nn_integral_cong[of M f g] nn_integral_cong[of "restrict_space M \<Omega>" f g] |
54417 | 1679 |
sets.sets_into_space[OF `\<Omega> \<in> sets M`] |
1680 |
by (simp add: subset_eq space_restrict_space) |
|
1681 |
next |
|
1682 |
case (set A) |
|
1683 |
then have "A \<subseteq> \<Omega>" |
|
1684 |
unfolding indicator_eq_0_iff by (auto dest: sets.sets_into_space) |
|
1685 |
with set `\<Omega> \<in> sets M` sets.sets_into_space[OF `\<Omega> \<in> sets M`] show ?case |
|
56996 | 1686 |
by (subst nn_integral_indicator') |
54417 | 1687 |
(auto simp add: sets_restrict_space_iff space_restrict_space |
1688 |
emeasure_restrict_space Int_absorb2 |
|
1689 |
dest: sets.sets_into_space) |
|
1690 |
next |
|
1691 |
case (mult f c) then show ?case |
|
56996 | 1692 |
by (cases "c = 0") (simp_all add: measurable_restrict_space1 \<Omega> nn_integral_cmult) |
54417 | 1693 |
next |
1694 |
case (add f g) then show ?case |
|
56996 | 1695 |
by (simp add: measurable_restrict_space1 \<Omega> nn_integral_add ereal_add_nonneg_eq_0_iff) |
54417 | 1696 |
next |
1697 |
case (seq F) then show ?case |
|
56996 | 1698 |
by (auto simp add: SUP_eq_iff measurable_restrict_space1 \<Omega> nn_integral_monotone_convergence_SUP) |
54417 | 1699 |
qed |
1700 |
||
56994 | 1701 |
subsubsection {* Measure spaces with an associated density *} |
47694 | 1702 |
|
1703 |
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
|
1704 |
"density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)" |
35582 | 1705 |
|
47694 | 1706 |
lemma |
1707 |
shows sets_density[simp]: "sets (density M f) = sets M" |
|
1708 |
and space_density[simp]: "space (density M f) = space M" |
|
1709 |
by (auto simp: density_def) |
|
1710 |
||
50003 | 1711 |
(* FIXME: add conversion to simplify space, sets and measurable *) |
1712 |
lemma space_density_imp[measurable_dest]: |
|
1713 |
"\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto |
|
1714 |
||
47694 | 1715 |
lemma |
1716 |
shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'" |
|
1717 |
and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'" |
|
1718 |
and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u" |
|
1719 |
unfolding measurable_def simple_function_def by simp_all |
|
1720 |
||
1721 |
lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow> |
|
1722 |
(AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'" |
|
56996 | 1723 |
unfolding density_def by (auto intro!: measure_of_eq nn_integral_cong_AE sets.space_closed) |
47694 | 1724 |
|
1725 |
lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))" |
|
1726 |
proof - |
|
1727 |
have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)" |
|
1728 |
by (auto simp: indicator_def) |
|
1729 |
then show ?thesis |
|
56996 | 1730 |
unfolding density_def by (simp add: nn_integral_max_0) |
47694 | 1731 |
qed |
1732 |
||
1733 |
lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))" |
|
1734 |
by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max) |
|
38656 | 1735 |
|
47694 | 1736 |
lemma emeasure_density: |
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1737 |
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
|
1738 |
shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)" |
47694 | 1739 |
(is "_ = ?\<mu> A") |
1740 |
unfolding density_def |
|
1741 |
proof (rule emeasure_measure_of_sigma) |
|
1742 |
show "sigma_algebra (space M) (sets M)" .. |
|
1743 |
show "positive (sets M) ?\<mu>" |
|
56996 | 1744 |
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
|
1745 |
have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'") |
56996 | 1746 |
apply (subst nn_integral_max_0[symmetric]) |
1747 |
apply (intro ext nn_integral_cong_AE AE_I2) |
|
47694 | 1748 |
apply (auto simp: indicator_def) |
1749 |
done |
|
1750 |
show "countably_additive (sets M) ?\<mu>" |
|
1751 |
unfolding \<mu>_eq |
|
1752 |
proof (intro countably_additiveI) |
|
1753 |
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
|
1754 |
then have "\<And>i. A i \<in> sets M" by auto |
47694 | 1755 |
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
|
1756 |
by (auto simp: set_eq_iff) |
47694 | 1757 |
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
|
1758 |
have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)" |
56996 | 1759 |
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
|
1760 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f |
56996 | 1761 |
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
|
1762 |
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)" |
47694 | 1763 |
unfolding suminf_indicator[OF disj] .. |
1764 |
finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp |
|
1765 |
qed |
|
1766 |
qed fact |
|
38656 | 1767 |
|
47694 | 1768 |
lemma null_sets_density_iff: |
1769 |
assumes f: "f \<in> borel_measurable M" |
|
1770 |
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)" |
|
1771 |
proof - |
|
1772 |
{ 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
|
1773 |
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 | 1774 |
apply (subst nn_integral_max_0[symmetric]) |
1775 |
apply (intro nn_integral_cong) |
|
47694 | 1776 |
apply (auto simp: indicator_def) |
1777 |
done |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1778 |
have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> |
47694 | 1779 |
emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0" |
1780 |
unfolding eq |
|
1781 |
using f `A \<in> sets M` |
|
56996 | 1782 |
by (intro nn_integral_0_iff) auto |
47694 | 1783 |
also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)" |
1784 |
using f `A \<in> sets M` |
|
50002
ce0d316b5b44
add measurability prover; add support for Borel sets
hoelzl
parents:
50001
diff
changeset
|
1785 |
by (intro AE_iff_measurable[OF _ refl, symmetric]) auto |
47694 | 1786 |
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)" |
1787 |
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
|
1788 |
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 | 1789 |
with f show ?thesis |
1790 |
by (simp add: null_sets_def emeasure_density cong: conj_cong) |
|
1791 |
qed |
|
1792 |
||
1793 |
lemma AE_density: |
|
1794 |
assumes f: "f \<in> borel_measurable M" |
|
1795 |
shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)" |
|
1796 |
proof |
|
1797 |
assume "AE x in density M f. P x" |
|
1798 |
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" |
|
1799 |
by (auto simp: eventually_ae_filter null_sets_density_iff) |
|
1800 |
then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto |
|
1801 |
with ae show "AE x in M. 0 < f x \<longrightarrow> P x" |
|
1802 |
by (rule eventually_elim2) auto |
|
1803 |
next |
|
1804 |
fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x" |
|
1805 |
then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M" |
|
1806 |
by (auto simp: eventually_ae_filter) |
|
1807 |
then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}" |
|
1808 |
"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
|
1809 |
using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in) |
47694 | 1810 |
show "AE x in density M f. P x" |
1811 |
using ae2 |
|
1812 |
unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f] |
|
1813 |
by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *) |
|
1814 |
(auto elim: eventually_elim2) |
|
35582 | 1815 |
qed |
1816 |
||
56996 | 1817 |
lemma nn_integral_density': |
47694 | 1818 |
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
|
1819 |
assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x" |
56996 | 1820 |
shows "integral\<^sup>N (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)" |
49798 | 1821 |
using g proof induct |
1822 |
case (cong u v) |
|
49799
15ea98537c76
strong nonnegativ (instead of ae nn) for induction rule
hoelzl
parents:
49798
diff
changeset
|
1823 |
then show ?case |
56996 | 1824 |
apply (subst nn_integral_cong[OF cong(3)]) |
1825 |
apply (simp_all cong: nn_integral_cong) |
|
49798 | 1826 |
done |
1827 |
next |
|
1828 |
case (set A) then show ?case |
|
1829 |
by (simp add: emeasure_density f) |
|
1830 |
next |
|
1831 |
case (mult u c) |
|
1832 |
moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps) |
|
1833 |
ultimately show ?case |
|
56996 | 1834 |
using f by (simp add: nn_integral_cmult) |
49798 | 1835 |
next |
1836 |
case (add u v) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1837 |
then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x" |
49798 | 1838 |
by (simp add: ereal_right_distrib) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53015
diff
changeset
|
1839 |
with add f show ?case |
56996 | 1840 |
by (auto simp add: nn_integral_add ereal_zero_le_0_iff intro!: nn_integral_add[symmetric]) |
49798 | 1841 |
next |
1842 |
case (seq U) |
|
1843 |
from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)" |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56193
diff
changeset
|
1844 |
by eventually_elim (simp add: SUP_ereal_cmult seq) |
49798 | 1845 |
from seq f show ?case |
56996 | 1846 |
apply (simp add: nn_integral_monotone_convergence_SUP) |
1847 |
apply (subst nn_integral_cong_AE[OF eq]) |
|
1848 |
apply (subst nn_integral_monotone_convergence_SUP_AE) |
|
49798 | 1849 |
apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono) |
1850 |
done |
|
47694 | 1851 |
qed |
38705 | 1852 |
|
56996 | 1853 |
lemma nn_integral_density: |
49798 | 1854 |
"f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow> |
56996 | 1855 |
integral\<^sup>N (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)" |
1856 |
by (subst (1 2) nn_integral_max_0[symmetric]) |
|
1857 |
(auto intro!: nn_integral_cong_AE |
|
1858 |
simp: measurable_If max_def ereal_zero_le_0_iff nn_integral_density') |
|
49798 | 1859 |
|
47694 | 1860 |
lemma emeasure_restricted: |
1861 |
assumes S: "S \<in> sets M" and X: "X \<in> sets M" |
|
1862 |
shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)" |
|
38705 | 1863 |
proof - |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
1864 |
have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)" |
47694 | 1865 |
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
|
1866 |
also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)" |
56996 | 1867 |
by (auto intro!: nn_integral_cong simp: indicator_def) |
47694 | 1868 |
also have "\<dots> = emeasure M (S \<inter> X)" |
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
1869 |
using S X by (simp add: sets.Int) |
47694 | 1870 |
finally show ?thesis . |
1871 |
qed |
|
1872 |
||
1873 |
lemma measure_restricted: |
|
1874 |
"S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)" |
|
1875 |
by (simp add: emeasure_restricted measure_def) |
|
1876 |
||
1877 |
lemma (in finite_measure) finite_measure_restricted: |
|
1878 |
"S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))" |
|
1879 |
by default (simp add: emeasure_restricted) |
|
1880 |
||
1881 |
lemma emeasure_density_const: |
|
1882 |
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A" |
|
56996 | 1883 |
by (auto simp: nn_integral_cmult_indicator emeasure_density) |
47694 | 1884 |
|
1885 |
lemma measure_density_const: |
|
1886 |
"A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A" |
|
1887 |
by (auto simp: emeasure_density_const measure_def) |
|
1888 |
||
1889 |
lemma density_density_eq: |
|
1890 |
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> |
|
1891 |
density (density M f) g = density M (\<lambda>x. f x * g x)" |
|
56996 | 1892 |
by (auto intro!: measure_eqI simp: emeasure_density nn_integral_density ac_simps) |
47694 | 1893 |
|
1894 |
lemma distr_density_distr: |
|
1895 |
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M" |
|
1896 |
and inv: "\<forall>x\<in>space M. T' (T x) = x" |
|
1897 |
assumes f: "f \<in> borel_measurable M'" |
|
1898 |
shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L") |
|
1899 |
proof (rule measure_eqI) |
|
1900 |
fix A assume A: "A \<in> sets ?R" |
|
1901 |
{ fix x assume "x \<in> space M" |
|
50244
de72bbe42190
qualified interpretation of sigma_algebra, to avoid name clashes
immler
parents:
50104
diff
changeset
|
1902 |
with sets.sets_into_space[OF A] |
47694 | 1903 |
have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)" |
1904 |
using T inv by (auto simp: indicator_def measurable_space) } |
|
1905 |
with A T T' f show "emeasure ?R A = emeasure ?L A" |
|
1906 |
by (simp add: measurable_comp emeasure_density emeasure_distr |
|
56996 | 1907 |
nn_integral_distr measurable_sets cong: nn_integral_cong) |
47694 | 1908 |
qed simp |
1909 |
||
1910 |
lemma density_density_divide: |
|
1911 |
fixes f g :: "'a \<Rightarrow> real" |
|
1912 |
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" |
|
1913 |
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x" |
|
1914 |
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0" |
|
1915 |
shows "density (density M f) (\<lambda>x. g x / f x) = density M g" |
|
1916 |
proof - |
|
1917 |
have "density M g = density M (\<lambda>x. f x * (g x / f x))" |
|
1918 |
using f g ac by (auto intro!: density_cong measurable_If) |
|
1919 |
then show ?thesis |
|
1920 |
using f g by (subst density_density_eq) auto |
|
38705 | 1921 |
qed |
1922 |
||
56994 | 1923 |
subsubsection {* Point measure *} |
47694 | 1924 |
|
1925 |
definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where |
|
1926 |
"point_measure A f = density (count_space A) f" |
|
1927 |
||
1928 |
lemma |
|
1929 |
shows space_point_measure: "space (point_measure A f) = A" |
|
1930 |
and sets_point_measure: "sets (point_measure A f) = Pow A" |
|
1931 |
by (auto simp: point_measure_def) |
|
1932 |
||
1933 |
lemma measurable_point_measure_eq1[simp]: |
|
1934 |
"g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M" |
|
1935 |
unfolding point_measure_def by simp |
|
1936 |
||
1937 |
lemma measurable_point_measure_eq2_finite[simp]: |
|
1938 |
"finite A \<Longrightarrow> |
|
1939 |
g \<in> measurable M (point_measure A f) \<longleftrightarrow> |
|
1940 |
(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
|
1941 |
unfolding point_measure_def by (simp add: measurable_count_space_eq2) |
47694 | 1942 |
|
1943 |
lemma simple_function_point_measure[simp]: |
|
1944 |
"simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)" |
|
1945 |
by (simp add: point_measure_def) |
|
1946 |
||
1947 |
lemma emeasure_point_measure: |
|
1948 |
assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A" |
|
1949 |
shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)" |
|
35977 | 1950 |
proof - |
47694 | 1951 |
have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}" |
1952 |
using `X \<subseteq> A` by auto |
|
1953 |
with A show ?thesis |
|
56996 | 1954 |
by (simp add: emeasure_density nn_integral_count_space ereal_zero_le_0_iff |
47694 | 1955 |
point_measure_def indicator_def) |
35977 | 1956 |
qed |
1957 |
||
47694 | 1958 |
lemma emeasure_point_measure_finite: |
49795 | 1959 |
"finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)" |
47694 | 1960 |
by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le) |
1961 |
||
49795 | 1962 |
lemma emeasure_point_measure_finite2: |
1963 |
"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)" |
|
1964 |
by (subst emeasure_point_measure) |
|
1965 |
(auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le) |
|
1966 |
||
47694 | 1967 |
lemma null_sets_point_measure_iff: |
1968 |
"X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)" |
|
1969 |
by (auto simp: AE_count_space null_sets_density_iff point_measure_def) |
|
1970 |
||
1971 |
lemma AE_point_measure: |
|
1972 |
"(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)" |
|
1973 |
unfolding point_measure_def |
|
1974 |
by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def) |
|
1975 |
||
56996 | 1976 |
lemma nn_integral_point_measure: |
47694 | 1977 |
"finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow> |
56996 | 1978 |
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 | 1979 |
unfolding point_measure_def |
1980 |
apply (subst density_max_0) |
|
56996 | 1981 |
apply (subst nn_integral_density) |
1982 |
apply (simp_all add: AE_count_space nn_integral_density) |
|
1983 |
apply (subst nn_integral_count_space ) |
|
47694 | 1984 |
apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff) |
1985 |
apply (rule finite_subset) |
|
1986 |
prefer 2 |
|
1987 |
apply assumption |
|
1988 |
apply auto |
|
1989 |
done |
|
1990 |
||
56996 | 1991 |
lemma nn_integral_point_measure_finite: |
47694 | 1992 |
"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 | 1993 |
integral\<^sup>N (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)" |
1994 |
by (subst nn_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le) |
|
47694 | 1995 |
|
56994 | 1996 |
subsubsection {* Uniform measure *} |
47694 | 1997 |
|
1998 |
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)" |
|
1999 |
||
2000 |
lemma |
|
2001 |
shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M" |
|
2002 |
and space_uniform_measure[simp]: "space (uniform_measure M A) = space M" |
|
2003 |
by (auto simp: uniform_measure_def) |
|
2004 |
||
2005 |
lemma emeasure_uniform_measure[simp]: |
|
2006 |
assumes A: "A \<in> sets M" and B: "B \<in> sets M" |
|
2007 |
shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A" |
|
2008 |
proof - |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51340
diff
changeset
|
2009 |
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 | 2010 |
by (auto simp add: uniform_measure_def emeasure_density split: split_indicator |
56996 | 2011 |
intro!: nn_integral_cong) |
47694 | 2012 |
also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A" |
2013 |
using A B |
|
56996 | 2014 |
by (subst nn_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg) |
47694 | 2015 |
finally show ?thesis . |
2016 |
qed |
|
2017 |
||
2018 |
lemma measure_uniform_measure[simp]: |
|
2019 |
assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M" |
|
2020 |
shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A" |
|
2021 |
using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A |
|
2022 |
by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def) |
|
2023 |
||
56994 | 2024 |
subsubsection {* Uniform count measure *} |
47694 | 2025 |
|
2026 |
definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)" |
|
2027 |
||
2028 |
lemma |
|
2029 |
shows space_uniform_count_measure: "space (uniform_count_measure A) = A" |
|
2030 |
and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A" |
|
2031 |
unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure) |
|
2032 |
||
2033 |
lemma emeasure_uniform_count_measure: |
|
2034 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A" |
|
2035 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def) |
|
2036 |
||
2037 |
lemma measure_uniform_count_measure: |
|
2038 |
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A" |
|
2039 |
by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def) |
|
2040 |
||
35748 | 2041 |
end |