author | eberlm <eberlm@in.tum.de> |
Thu, 30 Nov 2017 16:59:59 +0100 | |
changeset 67107 | cef76a19125e |
parent 66456 | 621897f47fab |
child 67339 | d91b9d22305b |
permissions | -rw-r--r-- |
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(* Title: HOL/Analysis/Set_Integral.thy |
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Author: Jeremy Avigad (CMU), Johannes Hölzl (TUM), Luke Serafin (CMU) |
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Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr |
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Notation and useful facts for working with integrals over a set. |
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|
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TODO: keep all these? Need unicode translations as well. |
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*) |
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|
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theory Set_Integral |
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imports Radon_Nikodym |
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begin |
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|
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Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
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(* |
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Notation |
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*) |
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|
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abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M" |
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|
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abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)" |
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|
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abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)" |
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|
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syntax |
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"_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real" |
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("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60) |
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translations |
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"LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)" |
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|
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(* |
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Notation for integration wrt lebesgue measure on the reals: |
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LBINT x. f |
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LBINT x : A. f |
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TODO: keep all these? Need unicode. |
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*) |
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syntax |
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"_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real" |
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("(2LBINT _./ _)" [0,60] 60) |
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syntax |
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"_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real" |
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("(3LBINT _:_./ _)" [0,60,61] 60) |
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(* |
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Basic properties |
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*) |
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(* |
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lemma indicator_abs_eq: "\<And>A x. \<bar>indicator A x\<bar> = ((indicator A x) :: real)" |
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by (auto simp add: indicator_def) |
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*) |
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lemma set_borel_measurable_sets: |
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fixes f :: "_ \<Rightarrow> _::real_normed_vector" |
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assumes "set_borel_measurable M X f" "B \<in> sets borel" "X \<in> sets M" |
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shows "f -` B \<inter> X \<in> sets M" |
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proof - |
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have "f \<in> borel_measurable (restrict_space M X)" |
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using assms by (subst borel_measurable_restrict_space_iff) auto |
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then have "f -` B \<inter> space (restrict_space M X) \<in> sets (restrict_space M X)" |
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by (rule measurable_sets) fact |
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with \<open>X \<in> sets M\<close> show ?thesis |
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by (subst (asm) sets_restrict_space_iff) (auto simp: space_restrict_space) |
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qed |
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||
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lemma set_lebesgue_integral_cong: |
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assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x" |
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shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)" |
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using assms by (auto intro!: Bochner_Integration.integral_cong split: split_indicator simp add: sets.sets_into_space) |
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lemma set_lebesgue_integral_cong_AE: |
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assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M" |
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assumes "AE x \<in> A in M. f x = g x" |
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shows "LINT x:A|M. f x = LINT x:A|M. g x" |
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proof- |
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have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x" |
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using assms by auto |
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thus ?thesis by (intro integral_cong_AE) auto |
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qed |
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|
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lemma set_integrable_cong_AE: |
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"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> |
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AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow> |
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set_integrable M A f = set_integrable M A g" |
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by (rule integrable_cong_AE) auto |
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90 |
|
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lemma set_integrable_subset: |
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fixes M A B and f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
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assumes "set_integrable M A f" "B \<in> sets M" "B \<subseteq> A" |
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shows "set_integrable M B f" |
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95 |
proof - |
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have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)" |
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by (rule integrable_mult_indicator) fact+ |
61808 | 98 |
with \<open>B \<subseteq> A\<close> show ?thesis |
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by (simp add: indicator_inter_arith[symmetric] Int_absorb2) |
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100 |
qed |
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101 |
|
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(* TODO: integral_cmul_indicator should be named set_integral_const *) |
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(* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *) |
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|
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lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)" |
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by (subst integral_scaleR_right[symmetric]) (auto intro!: Bochner_Integration.integral_cong) |
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107 |
|
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paulson <lp15@cam.ac.uk>
parents:
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108 |
lemma set_integral_mult_right [simp]: |
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fixes a :: "'a::{real_normed_field, second_countable_topology}" |
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shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)" |
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by (subst integral_mult_right_zero[symmetric]) auto |
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112 |
|
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paulson <lp15@cam.ac.uk>
parents:
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lemma set_integral_mult_left [simp]: |
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fixes a :: "'a::{real_normed_field, second_countable_topology}" |
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shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a" |
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by (subst integral_mult_left_zero[symmetric]) auto |
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117 |
|
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118 |
lemma set_integral_divide_zero [simp]: |
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haftmann
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59358
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|
119 |
fixes a :: "'a::{real_normed_field, field, second_countable_topology}" |
59092
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|
120 |
shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a" |
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685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
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121 |
by (subst integral_divide_zero[symmetric], intro Bochner_Integration.integral_cong) |
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122 |
(auto split: split_indicator) |
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123 |
|
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124 |
lemma set_integrable_scaleR_right [simp, intro]: |
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|
125 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)" |
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|
126 |
unfolding scaleR_left_commute by (rule integrable_scaleR_right) |
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|
127 |
|
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|
128 |
lemma set_integrable_scaleR_left [simp, intro]: |
d469103c0737
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129 |
fixes a :: "_ :: {banach, second_countable_topology}" |
d469103c0737
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diff
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|
130 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)" |
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131 |
using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
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|
132 |
|
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|
133 |
lemma set_integrable_mult_right [simp, intro]: |
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134 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
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|
135 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)" |
d469103c0737
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|
136 |
using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
d469103c0737
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|
137 |
|
d469103c0737
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138 |
lemma set_integrable_mult_left [simp, intro]: |
d469103c0737
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139 |
fixes a :: "'a::{real_normed_field, second_countable_topology}" |
d469103c0737
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diff
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|
140 |
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)" |
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|
141 |
using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp |
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|
142 |
|
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143 |
lemma set_integrable_divide [simp, intro]: |
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144 |
fixes a :: "'a::{real_normed_field, field, second_countable_topology}" |
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145 |
assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f" |
d469103c0737
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|
146 |
shows "set_integrable M A (\<lambda>t. f t / a)" |
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|
147 |
proof - |
d469103c0737
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148 |
have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)" |
d469103c0737
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149 |
using assms by (rule integrable_divide_zero) |
d469103c0737
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150 |
also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))" |
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151 |
by (auto split: split_indicator) |
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152 |
finally show ?thesis . |
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153 |
qed |
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154 |
|
d469103c0737
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155 |
lemma set_integral_add [simp, intro]: |
d469103c0737
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156 |
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
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157 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
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158 |
shows "set_integrable M A (\<lambda>x. f x + g x)" |
d469103c0737
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159 |
and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)" |
d469103c0737
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|
160 |
using assms by (simp_all add: scaleR_add_right) |
d469103c0737
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diff
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|
161 |
|
d469103c0737
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162 |
lemma set_integral_diff [simp, intro]: |
d469103c0737
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diff
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|
163 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
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diff
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164 |
shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x = |
d469103c0737
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|
165 |
(LINT x:A|M. f x) - (LINT x:A|M. g x)" |
d469103c0737
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diff
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|
166 |
using assms by (simp_all add: scaleR_diff_right) |
d469103c0737
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diff
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|
167 |
|
d469103c0737
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diff
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|
168 |
(* question: why do we have this for negation, but multiplication by a constant |
d469103c0737
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169 |
requires an integrability assumption? *) |
d469103c0737
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|
170 |
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)" |
d469103c0737
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diff
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|
171 |
by (subst integral_minus[symmetric]) simp_all |
d469103c0737
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|
172 |
|
d469103c0737
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diff
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|
173 |
lemma set_integral_complex_of_real: |
d469103c0737
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|
174 |
"LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)" |
d469103c0737
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diff
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|
175 |
by (subst integral_complex_of_real[symmetric]) |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
176 |
(auto intro!: Bochner_Integration.integral_cong split: split_indicator) |
59092
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|
177 |
|
d469103c0737
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diff
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|
178 |
lemma set_integral_mono: |
d469103c0737
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diff
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|
179 |
fixes f g :: "_ \<Rightarrow> real" |
d469103c0737
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diff
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|
180 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
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diff
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|
181 |
"\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x" |
d469103c0737
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diff
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|
182 |
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)" |
d469103c0737
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diff
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|
183 |
using assms by (auto intro: integral_mono split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
changeset
|
184 |
|
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
185 |
lemma set_integral_mono_AE: |
59092
d469103c0737
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186 |
fixes f g :: "_ \<Rightarrow> real" |
d469103c0737
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diff
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|
187 |
assumes "set_integrable M A f" "set_integrable M A g" |
d469103c0737
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diff
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|
188 |
"AE x \<in> A in M. f x \<le> g x" |
d469103c0737
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hoelzl
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diff
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|
189 |
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
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|
190 |
using assms by (auto intro: integral_mono_AE split: split_indicator) |
d469103c0737
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diff
changeset
|
191 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
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|
192 |
lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
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|
193 |
using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps) |
d469103c0737
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diff
changeset
|
194 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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diff
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|
195 |
lemma set_integrable_abs_iff: |
d469103c0737
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diff
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|
196 |
fixes f :: "_ \<Rightarrow> real" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
197 |
shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f" |
59092
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198 |
by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps) |
d469103c0737
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diff
changeset
|
199 |
|
d469103c0737
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diff
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|
200 |
lemma set_integrable_abs_iff': |
d469103c0737
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diff
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|
201 |
fixes f :: "_ \<Rightarrow> real" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
202 |
shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow> |
59092
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|
203 |
set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f" |
d469103c0737
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|
204 |
by (intro set_integrable_abs_iff) auto |
d469103c0737
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diff
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|
205 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
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206 |
lemma set_integrable_discrete_difference: |
d469103c0737
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diff
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|
207 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
d469103c0737
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208 |
assumes "countable X" |
d469103c0737
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|
209 |
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X" |
d469103c0737
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diff
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|
210 |
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" |
d469103c0737
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|
211 |
shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f" |
d469103c0737
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212 |
proof (rule integrable_discrete_difference[where X=X]) |
d469103c0737
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diff
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|
213 |
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x" |
d469103c0737
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|
214 |
using diff by (auto split: split_indicator) |
d469103c0737
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diff
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|
215 |
qed fact+ |
d469103c0737
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diff
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|
216 |
|
d469103c0737
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diff
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|
217 |
lemma set_integral_discrete_difference: |
d469103c0737
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|
218 |
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
219 |
assumes "countable X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
220 |
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
221 |
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
222 |
shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
223 |
proof (rule integral_discrete_difference[where X=X]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
224 |
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
225 |
using diff by (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
226 |
qed fact+ |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
227 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
228 |
lemma set_integrable_Un: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
229 |
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
230 |
assumes f_A: "set_integrable M A f" and f_B: "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
231 |
and [measurable]: "A \<in> sets M" "B \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
232 |
shows "set_integrable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
233 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
234 |
have "set_integrable M (A - B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
235 |
using f_A by (rule set_integrable_subset) auto |
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63627
diff
changeset
|
236 |
from Bochner_Integration.integrable_add[OF this f_B] show ?thesis |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
237 |
by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
238 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
239 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
240 |
lemma set_integrable_UN: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
241 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
242 |
assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
243 |
"\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
244 |
shows "set_integrable M (\<Union>i\<in>I. A i) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
245 |
using assms by (induct I) (auto intro!: set_integrable_Un) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
246 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
247 |
lemma set_integral_Un: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
248 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
249 |
assumes "A \<inter> B = {}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
250 |
and "set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
251 |
and "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
252 |
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
253 |
by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric] |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
254 |
scaleR_add_left assms) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
255 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
256 |
lemma set_integral_cong_set: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
257 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
258 |
assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
259 |
and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
260 |
shows "LINT x:B|M. f x = LINT x:A|M. f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
261 |
proof (rule integral_cong_AE) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
262 |
show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
263 |
using ae by (auto simp: subset_eq split: split_indicator) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
264 |
qed fact+ |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
265 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
266 |
lemma set_borel_measurable_subset: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
267 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
268 |
assumes [measurable]: "set_borel_measurable M A f" "B \<in> sets M" and "B \<subseteq> A" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
269 |
shows "set_borel_measurable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
270 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
271 |
have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
272 |
by measurable |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
273 |
also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)" |
61808 | 274 |
using \<open>B \<subseteq> A\<close> by (auto simp: fun_eq_iff split: split_indicator) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
275 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
276 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
277 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
278 |
lemma set_integral_Un_AE: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
279 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
280 |
assumes ae: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)" and [measurable]: "A \<in> sets M" "B \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
281 |
and "set_integrable M A f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
282 |
and "set_integrable M B f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
283 |
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
284 |
proof - |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
285 |
have f: "set_integrable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
286 |
by (intro set_integrable_Un assms) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
287 |
then have f': "set_borel_measurable M (A \<union> B) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
288 |
by (rule borel_measurable_integrable) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
289 |
have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x" |
60615
e5fa1d5d3952
Useful lemmas. The theorem concerning swapping the variables in a double integral.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
290 |
proof (rule set_integral_cong_set) |
59092
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
291 |
show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
292 |
using ae by auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
293 |
show "set_borel_measurable M (A - A \<inter> B \<union> (B - A \<inter> B)) f" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
294 |
using f' by (rule set_borel_measurable_subset) auto |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
295 |
qed fact |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
296 |
also have "\<dots> = (LINT x:(A - A \<inter> B)|M. f x) + (LINT x:(B - A \<inter> B)|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
297 |
by (auto intro!: set_integral_Un set_integrable_subset[OF f]) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
298 |
also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
299 |
using ae |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
300 |
by (intro arg_cong2[where f="op+"] set_integral_cong_set) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
301 |
(auto intro!: set_borel_measurable_subset[OF f']) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
302 |
finally show ?thesis . |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
303 |
qed |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
304 |
|
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
305 |
lemma set_integral_finite_Union: |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
306 |
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
307 |
assumes "finite I" "disjoint_family_on A I" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
308 |
and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
309 |
shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)" |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
310 |
using assms |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
311 |
apply induct |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
312 |
apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
diff
changeset
|
313 |
by (subst set_integral_Un, auto intro: set_integrable_UN) |
d469103c0737
add integral substitution theorems from Manuel Eberl, Jeremy Avigad, Luke Serafin, and Sudeep Kanav
hoelzl
parents:
|