(* Title: HOL/Probability/Nonnegative_Lebesgue_Integration.thy
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
section \<open>Lebesgue Integration for Nonnegative Functions\<close>
theory Nonnegative_Lebesgue_Integration
imports Measure_Space Borel_Space
begin
lemma infinite_countable_subset':
assumes X: "infinite X" shows "\<exists>C\<subseteq>X. countable C \<and> infinite C"
proof -
from infinite_countable_subset[OF X] guess f ..
then show ?thesis
by (intro exI[of _ "range f"]) (auto simp: range_inj_infinite)
qed
lemma indicator_less_ereal[simp]:
"indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
by (simp add: indicator_def not_le)
subsection "Simple function"
text \<open>
Our simple functions are not restricted to nonnegative real numbers. Instead
they are just functions with a finite range and are measurable when singleton
sets are measurable.
\<close>
definition "simple_function M g \<longleftrightarrow>
finite (g ` space M) \<and>
(\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
lemma simple_functionD:
assumes "simple_function M g"
shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
proof -
show "finite (g ` space M)"
using assms unfolding simple_function_def by auto
have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
finally show "g -` X \<inter> space M \<in> sets M" using assms
by (auto simp del: UN_simps simp: simple_function_def)
qed
lemma measurable_simple_function[measurable_dest]:
"simple_function M f \<Longrightarrow> f \<in> measurable M (count_space UNIV)"
unfolding simple_function_def measurable_def
proof safe
fix A assume "finite (f ` space M)" "\<forall>x\<in>f ` space M. f -` {x} \<inter> space M \<in> sets M"
then have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) \<in> sets M"
by (intro sets.finite_UN) auto
also have "(\<Union>x\<in>f ` space M. if x \<in> A then f -` {x} \<inter> space M else {}) = f -` A \<inter> space M"
by (auto split: split_if_asm)
finally show "f -` A \<inter> space M \<in> sets M" .
qed simp
lemma borel_measurable_simple_function:
"simple_function M f \<Longrightarrow> f \<in> borel_measurable M"
by (auto dest!: measurable_simple_function simp: measurable_def)
lemma simple_function_measurable2[intro]:
assumes "simple_function M f" "simple_function M g"
shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
proof -
have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
by auto
then show ?thesis using assms[THEN simple_functionD(2)] by auto
qed
lemma simple_function_indicator_representation:
fixes f ::"'a \<Rightarrow> ereal"
assumes f: "simple_function M f" and x: "x \<in> space M"
shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
(is "?l = ?r")
proof -
have "?r = (\<Sum>y \<in> f ` space M.
(if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
by (auto intro!: setsum.cong)
also have "... = f x * indicator (f -` {f x} \<inter> space M) x"
using assms by (auto dest: simple_functionD simp: setsum.delta)
also have "... = f x" using x by (auto simp: indicator_def)
finally show ?thesis by auto
qed
lemma simple_function_notspace:
"simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
proof -
have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
have "?h -` {0} \<inter> space M = space M" by auto
thus ?thesis unfolding simple_function_def by auto
qed
lemma simple_function_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "simple_function M f \<longleftrightarrow> simple_function M g"
proof -
have "f ` space M = g ` space M"
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
using assms by (auto intro!: image_eqI)
thus ?thesis unfolding simple_function_def using assms by simp
qed
lemma simple_function_cong_algebra:
assumes "sets N = sets M" "space N = space M"
shows "simple_function M f \<longleftrightarrow> simple_function N f"
unfolding simple_function_def assms ..
lemma simple_function_borel_measurable:
fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
shows "simple_function M f"
using assms unfolding simple_function_def
by (auto intro: borel_measurable_vimage)
lemma simple_function_eq_measurable:
fixes f :: "'a \<Rightarrow> ereal"
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> measurable M (count_space UNIV)"
using simple_function_borel_measurable[of f] measurable_simple_function[of M f]
by (fastforce simp: simple_function_def)
lemma simple_function_const[intro, simp]:
"simple_function M (\<lambda>x. c)"
by (auto intro: finite_subset simp: simple_function_def)
lemma simple_function_compose[intro, simp]:
assumes "simple_function M f"
shows "simple_function M (g \<circ> f)"
unfolding simple_function_def
proof safe
show "finite ((g \<circ> f) ` space M)"
using assms unfolding simple_function_def by (auto simp: image_comp [symmetric])
next
fix x assume "x \<in> space M"
let ?G = "g -` {g (f x)} \<inter> (f`space M)"
have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
(\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
using assms unfolding simple_function_def *
by (rule_tac sets.finite_UN) auto
qed
lemma simple_function_indicator[intro, simp]:
assumes "A \<in> sets M"
shows "simple_function M (indicator A)"
proof -
have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
by (auto simp: indicator_def)
hence "finite ?S" by (rule finite_subset) simp
moreover have "- A \<inter> space M = space M - A" by auto
ultimately show ?thesis unfolding simple_function_def
using assms by (auto simp: indicator_def [abs_def])
qed
lemma simple_function_Pair[intro, simp]:
assumes "simple_function M f"
assumes "simple_function M g"
shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
unfolding simple_function_def
proof safe
show "finite (?p ` space M)"
using assms unfolding simple_function_def
by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
next
fix x assume "x \<in> space M"
have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
(f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
by auto
with \<open>x \<in> space M\<close> show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
using assms unfolding simple_function_def by auto
qed
lemma simple_function_compose1:
assumes "simple_function M f"
shows "simple_function M (\<lambda>x. g (f x))"
using simple_function_compose[OF assms, of g]
by (simp add: comp_def)
lemma simple_function_compose2:
assumes "simple_function M f" and "simple_function M g"
shows "simple_function M (\<lambda>x. h (f x) (g x))"
proof -
have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
using assms by auto
thus ?thesis by (simp_all add: comp_def)
qed
lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
lemma simple_function_setsum[intro, simp]:
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
proof cases
assume "finite P" from this assms show ?thesis by induct auto
qed auto
lemma simple_function_ereal[intro, simp]:
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
shows "simple_function M (\<lambda>x. ereal (f x))"
by (rule simple_function_compose1[OF sf])
lemma simple_function_real_of_nat[intro, simp]:
fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
shows "simple_function M (\<lambda>x. real (f x))"
by (rule simple_function_compose1[OF sf])
lemma borel_measurable_implies_simple_function_sequence:
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "u \<in> borel_measurable M"
shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
(\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
proof -
def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else nat(floor (real_of_ereal (u x) * 2 ^ i))"
{ fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
proof (split split_if, intro conjI impI)
assume "\<not> real j \<le> u x"
then have "nat(floor (real_of_ereal (u x) * 2 ^ j)) \<le> nat(floor (j * 2 ^ j))"
by (cases "u x") (auto intro!: nat_mono floor_mono)
moreover have "real (nat(floor (j * 2 ^ j))) \<le> j * 2^j"
by linarith
ultimately show "nat(floor (real_of_ereal (u x) * 2 ^ j)) \<le> j * 2 ^ j"
unfolding of_nat_le_iff by auto
qed auto }
note f_upper = this
have real_f:
"\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (nat(floor (real_of_ereal (u x) * 2 ^ i))))"
unfolding f_def by auto
let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
show ?thesis
proof (intro exI[of _ ?g] conjI allI ballI)
fix i
have "simple_function M (\<lambda>x. real (f x i))"
proof (intro simple_function_borel_measurable)
show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
using u by (auto simp: real_f)
have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
using f_upper[of _ i] by auto
then show "finite ((\<lambda>x. real (f x i))`space M)"
by (rule finite_subset) auto
qed
then show "simple_function M (?g i)"
by (auto)
next
show "incseq ?g"
proof (intro incseq_ereal incseq_SucI le_funI)
fix x and i :: nat
have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
proof ((split split_if)+, intro conjI impI)
assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
then show "i * 2 ^ i * 2 \<le> nat(floor (real_of_ereal (u x) * 2 ^ Suc i))"
by (cases "u x") (auto intro!: le_nat_floor)
next
assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
then show "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 \<le> Suc i * 2 ^ Suc i"
by (cases "u x") auto
next
assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
have "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 = nat(floor (real_of_ereal (u x) * 2 ^ i)) * nat(floor(2::real))"
by simp
also have "\<dots> \<le> nat(floor (real_of_ereal (u x) * 2 ^ i * 2))"
proof cases
assume "0 \<le> u x" then show ?thesis
by (intro le_mult_nat_floor)
next
assume "\<not> 0 \<le> u x" then show ?thesis
by (cases "u x") (auto simp: nat_floor_neg mult_nonpos_nonneg)
qed
also have "\<dots> = nat(floor (real_of_ereal (u x) * 2 ^ Suc i))"
by (simp add: ac_simps)
finally show "nat(floor (real_of_ereal (u x) * 2 ^ i)) * 2 \<le> nat(floor (real_of_ereal (u x) * 2 ^ Suc i))" .
qed simp
then show "?g i x \<le> ?g (Suc i) x"
by (auto simp: field_simps)
qed
next
fix x show "(SUP i. ?g i x) = max 0 (u x)"
proof (rule SUP_eqI)
fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
by (cases "u x") (auto simp: field_simps nat_floor_neg mult_nonpos_nonneg)
next
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
have "\<And>i. 0 \<le> ?g i x" by auto
from order_trans[OF this *] have "0 \<le> y" by simp
show "max 0 (u x) \<le> y"
proof (cases y)
case (real r)
with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
then guess p .. note ux = this
obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
have "p \<le> r"
proof (rule ccontr)
assume "\<not> p \<le> r"
with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: field_simps)
then have "r * 2^max N m < p * 2^max N m - 1" by simp
moreover
have "real (nat(floor (p * 2 ^ max N m))) \<le> r * 2 ^ max N m"
using *[of "max N m"] m unfolding real_f using ux
by (cases "0 \<le> u x") (simp_all add: max_def split: split_if_asm)
then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
by linarith
ultimately show False by auto
qed
then show "max 0 (u x) \<le> y" using real ux by simp
qed (insert \<open>0 \<le> y\<close>, auto)
qed
qed auto
qed
lemma borel_measurable_implies_simple_function_sequence':
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "u \<in> borel_measurable M"
obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
"\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
using borel_measurable_implies_simple_function_sequence[OF u] by auto
lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "simple_function M u"
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"
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
shows "P u"
proof (rule cong)
from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
proof eventually_elim
fix x assume x: "x \<in> space M"
from simple_function_indicator_representation[OF u x]
show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
qed
next
from u have "finite (u ` space M)"
unfolding simple_function_def by auto
then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
proof induct
case empty show ?case
using set[of "{}"] by (simp add: indicator_def[abs_def])
qed (auto intro!: add mult set simple_functionD u)
next
show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
apply (subst simple_function_cong)
apply (rule simple_function_indicator_representation[symmetric])
apply (auto intro: u)
done
qed fact
lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
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"
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
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)"
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)"
shows "P u"
proof -
show ?thesis
proof (rule cong)
fix x assume x: "x \<in> space M"
from simple_function_indicator_representation[OF u x]
show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
next
show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
apply (subst simple_function_cong)
apply (rule simple_function_indicator_representation[symmetric])
apply (auto intro: u)
done
next
from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
unfolding simple_function_def by auto
then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
proof induct
case empty show ?case
using set[of "{}"] by (simp add: indicator_def[abs_def])
next
case (insert x S)
{ fix z have "(\<Sum>y\<in>S. y * indicator (u -` {y} \<inter> space M) z) = 0 \<or>
x * indicator (u -` {x} \<inter> space M) z = 0"
using insert by (subst setsum_ereal_0) (auto simp: indicator_def) }
note disj = this
from insert show ?case
by (auto intro!: add mult set simple_functionD u setsum_nonneg simple_function_setsum disj)
qed
qed fact
qed
lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
fixes u :: "'a \<Rightarrow> ereal"
assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
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"
assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
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)"
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)"
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)"
shows "P u"
using u
proof (induct rule: borel_measurable_implies_simple_function_sequence')
fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
have u_eq: "u = (SUP i. U i)"
using nn u sup by (auto simp: max_def)
have not_inf: "\<And>x i. x \<in> space M \<Longrightarrow> U i x < \<infinity>"
using U by (auto simp: image_iff eq_commute)
from U have "\<And>i. U i \<in> borel_measurable M"
by (simp add: borel_measurable_simple_function)
show "P u"
unfolding u_eq
proof (rule seq)
fix i show "P (U i)"
using \<open>simple_function M (U i)\<close> nn[of i] not_inf[of _ i]
proof (induct rule: simple_function_induct_nn)
case (mult u c)
show ?case
proof cases
assume "c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0)"
with mult(2) show ?thesis
by (intro cong[of "\<lambda>x. c * u x" "indicator {}"] set)
(auto dest!: borel_measurable_simple_function)
next
assume "\<not> (c = 0 \<or> space M = {} \<or> (\<forall>x\<in>space M. u x = 0))"
with mult obtain x where u_fin: "\<And>x. x \<in> space M \<Longrightarrow> u x < \<infinity>"
and x: "x \<in> space M" "u x \<noteq> 0" "c \<noteq> 0"
by auto
with mult have "P u"
by auto
from x mult(5)[OF \<open>x \<in> space M\<close>] mult(1) mult(3)[of x] have "c < \<infinity>"
by auto
with u_fin mult
show ?thesis
by (intro mult') (auto dest!: borel_measurable_simple_function)
qed
qed (auto intro: cong intro!: set add dest!: borel_measurable_simple_function)
qed fact+
qed
lemma simple_function_If_set:
assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
proof -
def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
show ?thesis unfolding simple_function_def
proof safe
have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
from finite_subset[OF this] assms
show "finite (?IF ` space M)" unfolding simple_function_def by auto
next
fix x assume "x \<in> space M"
then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
using sets.sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
qed
qed
lemma simple_function_If:
assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
shows "simple_function M (\<lambda>x. if P x then f x else g x)"
proof -
have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
qed
lemma simple_function_subalgebra:
assumes "simple_function N f"
and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
shows "simple_function M f"
using assms unfolding simple_function_def by auto
lemma simple_function_comp:
assumes T: "T \<in> measurable M M'"
and f: "simple_function M' f"
shows "simple_function M (\<lambda>x. f (T x))"
proof (intro simple_function_def[THEN iffD2] conjI ballI)
have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
using T unfolding measurable_def by auto
then show "finite ((\<lambda>x. f (T x)) ` space M)"
using f unfolding simple_function_def by (auto intro: finite_subset)
fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
then have "i \<in> f ` space M'"
using T unfolding measurable_def by auto
then have "f -` {i} \<inter> space M' \<in> sets M'"
using f unfolding simple_function_def by auto
then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
using T unfolding measurable_def by auto
also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
using T unfolding measurable_def by auto
finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
qed
subsection "Simple integral"
definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>S") where
"integral\<^sup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
syntax
"_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>S _. _ \<partial>_" [60,61] 110)
translations
"\<integral>\<^sup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
lemma simple_integral_cong:
assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "integral\<^sup>S M f = integral\<^sup>S M g"
proof -
have "f ` space M = g ` space M"
"\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
using assms by (auto intro!: image_eqI)
thus ?thesis unfolding simple_integral_def by simp
qed
lemma simple_integral_const[simp]:
"(\<integral>\<^sup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
proof (cases "space M = {}")
case True thus ?thesis unfolding simple_integral_def by simp
next
case False hence "(\<lambda>x. c) ` space M = {c}" by auto
thus ?thesis unfolding simple_integral_def by simp
qed
lemma simple_function_partition:
assumes f: "simple_function M f" and g: "simple_function M g"
assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
shows "integral\<^sup>S M f = (\<Sum>y\<in>g ` space M. v y * emeasure M {x\<in>space M. g x = y})"
(is "_ = ?r")
proof -
from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
by (auto simp: simple_function_def)
from f g have [measurable]: "f \<in> measurable M (count_space UNIV)" "g \<in> measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function)
{ fix y assume "y \<in> space M"
then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
by (auto cong: sub simp: v[symmetric]) }
note eq = this
have "integral\<^sup>S M f =
(\<Sum>y\<in>f`space M. y * (\<Sum>z\<in>g`space M.
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))"
unfolding simple_integral_def
proof (safe intro!: setsum.cong ereal_right_mult_cong)
fix y assume y: "y \<in> space M" "f y \<noteq> 0"
have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} =
{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
by auto
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}) =
f -` {f y} \<inter> space M"
by (auto simp: eq_commute cong: sub rev_conj_cong)
have "finite (g`space M)" by simp
then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
by (rule rev_finite_subset) auto
then show "emeasure M (f -` {f y} \<inter> space M) =
(\<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)"
apply (simp add: setsum.If_cases)
apply (subst setsum_emeasure)
apply (auto simp: disjoint_family_on_def eq)
done
qed
also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
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))"
by (auto intro!: setsum.cong simp: setsum_ereal_right_distrib emeasure_nonneg)
also have "\<dots> = ?r"
by (subst setsum.commute)
(auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq)
finally show "integral\<^sup>S M f = ?r" .
qed
lemma simple_integral_add[simp]:
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"
shows "(\<integral>\<^sup>Sx. f x + g x \<partial>M) = integral\<^sup>S M f + integral\<^sup>S M g"
proof -
have "(\<integral>\<^sup>Sx. f x + g x \<partial>M) =
(\<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})"
by (intro simple_function_partition) (auto intro: f g)
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}) +
(\<Sum>y\<in>(\<lambda>x. (f x, g x))`space M. snd y * emeasure M {x\<in>space M. (f x, g x) = y})"
using assms(2,4) by (auto intro!: setsum.cong ereal_left_distrib simp: setsum.distrib[symmetric])
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)"
by (intro simple_function_partition[symmetric]) (auto intro: f g)
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)"
by (intro simple_function_partition[symmetric]) (auto intro: f g)
finally show ?thesis .
qed
lemma simple_integral_setsum[simp]:
assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
shows "(\<integral>\<^sup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>S M (f i))"
proof cases
assume "finite P"
from this assms show ?thesis
by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
qed auto
lemma simple_integral_mult[simp]:
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
shows "(\<integral>\<^sup>Sx. c * f x \<partial>M) = c * integral\<^sup>S M f"
proof -
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})"
using f by (intro simple_function_partition) auto
also have "\<dots> = c * integral\<^sup>S M f"
using f unfolding simple_integral_def
by (subst setsum_ereal_right_distrib) (auto simp: emeasure_nonneg mult.assoc Int_def conj_commute)
finally show ?thesis .
qed
lemma simple_integral_mono_AE:
assumes f[measurable]: "simple_function M f" and g[measurable]: "simple_function M g"
and mono: "AE x in M. f x \<le> g x"
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
proof -
let ?\<mu> = "\<lambda>P. emeasure M {x\<in>space M. P x}"
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))"
using f g by (intro simple_function_partition) auto
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))"
proof (clarsimp intro!: setsum_mono)
fix x assume "x \<in> space M"
let ?M = "?\<mu> (\<lambda>y. f y = f x \<and> g y = g x)"
show "f x * ?M \<le> g x * ?M"
proof cases
assume "?M \<noteq> 0"
then have "0 < ?M"
by (simp add: less_le emeasure_nonneg)
also have "\<dots> \<le> ?\<mu> (\<lambda>y. f x \<le> g x)"
using mono by (intro emeasure_mono_AE) auto
finally have "\<not> \<not> f x \<le> g x"
by (intro notI) auto
then show ?thesis
by (intro ereal_mult_right_mono) auto
qed simp
qed
also have "\<dots> = integral\<^sup>S M g"
using f g by (intro simple_function_partition[symmetric]) auto
finally show ?thesis .
qed
lemma simple_integral_mono:
assumes "simple_function M f" and "simple_function M g"
and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
shows "integral\<^sup>S M f \<le> integral\<^sup>S M g"
using assms by (intro simple_integral_mono_AE) auto
lemma simple_integral_cong_AE:
assumes "simple_function M f" and "simple_function M g"
and "AE x in M. f x = g x"
shows "integral\<^sup>S M f = integral\<^sup>S M g"
using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
lemma simple_integral_cong':
assumes sf: "simple_function M f" "simple_function M g"
and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
shows "integral\<^sup>S M f = integral\<^sup>S M g"
proof (intro simple_integral_cong_AE sf AE_I)
show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
using sf[THEN borel_measurable_simple_function] by auto
qed simp
lemma simple_integral_indicator:
assumes A: "A \<in> sets M"
assumes f: "simple_function M f"
shows "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
(\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
proof -
have eq: "(\<lambda>x. (f x, indicator A x)) ` space M \<inter> {x. snd x = 1} = (\<lambda>x. (f x, 1::ereal))`A"
using A[THEN sets.sets_into_space] by (auto simp: indicator_def image_iff split: split_if_asm)
have eq2: "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
by (auto simp: image_iff)
have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
(\<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})"
using assms by (intro simple_function_partition) auto
also have "\<dots> = (\<Sum>y\<in>(\<lambda>x. (f x, indicator A x::ereal))`space M.
if snd y = 1 then fst y * emeasure M (f -` {fst y} \<inter> space M \<inter> A) else 0)"
by (auto simp: indicator_def split: split_if_asm intro!: arg_cong2[where f="op *"] arg_cong2[where f=emeasure] setsum.cong)
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))"
using assms by (subst setsum.If_cases) (auto intro!: simple_functionD(1) simp: eq)
also have "\<dots> = (\<Sum>y\<in>fst`(\<lambda>x. (f x, 1::ereal))`A. y * emeasure M (f -` {y} \<inter> space M \<inter> A))"
by (subst setsum.reindex [of fst]) (auto simp: inj_on_def)
also have "\<dots> = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M \<inter> A))"
using A[THEN sets.sets_into_space]
by (intro setsum.mono_neutral_cong_left simple_functionD f) (auto simp: image_comp comp_def eq2)
finally show ?thesis .
qed
lemma simple_integral_indicator_only[simp]:
assumes "A \<in> sets M"
shows "integral\<^sup>S M (indicator A) = emeasure M A"
using simple_integral_indicator[OF assms, of "\<lambda>x. 1"] sets.sets_into_space[OF assms]
by (simp_all add: image_constant_conv Int_absorb1 split: split_if_asm)
lemma simple_integral_null_set:
assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
shows "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = 0"
proof -
have "AE x in M. indicator N x = (0 :: ereal)"
using \<open>N \<in> null_sets M\<close> by (auto simp: indicator_def intro!: AE_I[of _ _ N])
then have "(\<integral>\<^sup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^sup>Sx. 0 \<partial>M)"
using assms apply (intro simple_integral_cong_AE) by auto
then show ?thesis by simp
qed
lemma simple_integral_cong_AE_mult_indicator:
assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
shows "integral\<^sup>S M f = (\<integral>\<^sup>Sx. f x * indicator S x \<partial>M)"
using assms by (intro simple_integral_cong_AE) auto
lemma simple_integral_cmult_indicator:
assumes A: "A \<in> sets M"
shows "(\<integral>\<^sup>Sx. c * indicator A x \<partial>M) = c * emeasure M A"
using simple_integral_mult[OF simple_function_indicator[OF A]]
unfolding simple_integral_indicator_only[OF A] by simp
lemma simple_integral_nonneg:
assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
shows "0 \<le> integral\<^sup>S M f"
proof -
have "integral\<^sup>S M (\<lambda>x. 0) \<le> integral\<^sup>S M f"
using simple_integral_mono_AE[OF _ f ae] by auto
then show ?thesis by simp
qed
subsection \<open>Integral on nonnegative functions\<close>
definition nn_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>N") where
"integral\<^sup>N M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
syntax
"_nn_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+((2 _./ _)/ \<partial>_)" [60,61] 110)
translations
"\<integral>\<^sup>+x. f \<partial>M" == "CONST nn_integral M (\<lambda>x. f)"
lemma nn_integral_nonneg: "0 \<le> integral\<^sup>N M f"
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: nn_integral_def le_fun_def)
lemma nn_integral_le_0[simp]: "integral\<^sup>N M f \<le> 0 \<longleftrightarrow> integral\<^sup>N M f = 0"
using nn_integral_nonneg[of M f] by auto
lemma nn_integral_not_less_0 [simp]: "\<not> nn_integral M f < 0"
by(simp add: not_less nn_integral_nonneg)
lemma nn_integral_not_MInfty[simp]: "integral\<^sup>N M f \<noteq> -\<infinity>"
using nn_integral_nonneg[of M f] by auto
lemma nn_integral_def_finite:
"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)"
(is "_ = SUPREMUM ?A ?f")
unfolding nn_integral_def
proof (safe intro!: antisym SUP_least)
fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
note gM = g(1)[THEN borel_measurable_simple_function]
have \<mu>_G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
apply (safe intro!: simple_function_max simple_function_If)
apply (force simp: max_def le_fun_def split: split_if_asm)+
done
show "integral\<^sup>S M g \<le> SUPREMUM ?A ?f"
proof cases
have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
assume "(emeasure M) ?G = 0"
with gM have "AE x in M. x \<notin> ?G"
by (auto simp add: AE_iff_null intro!: null_setsI)
with gM g show ?thesis
by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
(auto simp: max_def intro!: simple_function_If)
next
assume \<mu>_G: "(emeasure M) ?G \<noteq> 0"
have "SUPREMUM ?A (integral\<^sup>S M) = \<infinity>"
proof (intro SUP_PInfty)
fix n :: nat
let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>_G \<mu>_G_pos by (auto simp: ereal_divide_eq)
then have "?g ?y \<in> ?A" by (rule g_in_A)
have "real n \<le> ?y * (emeasure M) ?G"
using \<mu>_G \<mu>_G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
also have "\<dots> = (\<integral>\<^sup>Sx. ?y * indicator ?G x \<partial>M)"
using \<open>0 \<le> ?y\<close> \<open>?g ?y \<in> ?A\<close> gM
by (subst simple_integral_cmult_indicator) auto
also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using \<open>?g ?y \<in> ?A\<close> gM
by (intro simple_integral_mono) auto
finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
using \<open>?g ?y \<in> ?A\<close> by blast
qed
then show ?thesis by simp
qed
qed (auto intro: SUP_upper)
lemma nn_integral_mono_AE:
assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
unfolding nn_integral_def
proof (safe intro!: SUP_mono)
fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
from ae[THEN AE_E] guess N . note N = this
then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
let ?n = "\<lambda>x. n x * indicator (space M - N) x"
have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
using n N ae_N by auto
moreover
{ fix x have "?n x \<le> max 0 (v x)"
proof cases
assume x: "x \<in> space M - N"
with N have "u x \<le> v x" by auto
with n(2)[THEN le_funD, of x] x show ?thesis
by (auto simp: max_def split: split_if_asm)
qed simp }
then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
moreover have "integral\<^sup>S M n \<le> integral\<^sup>S M ?n"
using ae_N N n by (auto intro!: simple_integral_mono_AE)
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"
by force
qed
lemma nn_integral_mono:
"(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>N M u \<le> integral\<^sup>N M v"
by (auto intro: nn_integral_mono_AE)
lemma mono_nn_integral: "mono F \<Longrightarrow> mono (\<lambda>x. integral\<^sup>N M (F x))"
by (auto simp add: mono_def le_fun_def intro!: nn_integral_mono)
lemma nn_integral_cong_AE:
"AE x in M. u x = v x \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v"
by (auto simp: eq_iff intro!: nn_integral_mono_AE)
lemma nn_integral_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v"
by (auto intro: nn_integral_cong_AE)
lemma nn_integral_cong_simp:
"(\<And>x. x \<in> space M =simp=> u x = v x) \<Longrightarrow> integral\<^sup>N M u = integral\<^sup>N M v"
by (auto intro: nn_integral_cong simp: simp_implies_def)
lemma nn_integral_cong_strong:
"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"
by (auto intro: nn_integral_cong)
lemma nn_integral_eq_simple_integral:
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>N M f = integral\<^sup>S M f"
proof -
let ?f = "\<lambda>x. f x * indicator (space M) x"
have f': "simple_function M ?f" using f by auto
with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
by (auto simp: fun_eq_iff max_def split: split_indicator)
have "integral\<^sup>N M ?f \<le> integral\<^sup>S M ?f" using f'
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def nn_integral_def)
moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>N M ?f"
unfolding nn_integral_def
using f' by (auto intro!: SUP_upper)
ultimately show ?thesis
by (simp cong: nn_integral_cong simple_integral_cong)
qed
lemma nn_integral_eq_simple_integral_AE:
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"
proof -
have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
with f have "integral\<^sup>N M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
by (simp cong: nn_integral_cong_AE simple_integral_cong_AE
add: nn_integral_eq_simple_integral)
with assms show ?thesis
by (auto intro!: simple_integral_cong_AE split: split_max)
qed
lemma nn_integral_SUP_approx:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
shows "integral\<^sup>S M u \<le> (SUP i. integral\<^sup>N M (f i))" (is "_ \<le> ?S")
proof (rule ereal_le_mult_one_interval)
have "0 \<le> (SUP i. integral\<^sup>N M (f i))"
using f(3) by (auto intro!: SUP_upper2 nn_integral_nonneg)
then show "(SUP i. integral\<^sup>N M (f i)) \<noteq> -\<infinity>" by auto
have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
using u(3) by auto
fix a :: ereal assume "0 < a" "a < 1"
hence "a \<noteq> 0" by auto
let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
have B: "\<And>i. ?B i \<in> sets M"
using f \<open>simple_function M u\<close>[THEN borel_measurable_simple_function] by auto
let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
{ fix i have "?B i \<subseteq> ?B (Suc i)"
proof safe
fix i x assume "a * u x \<le> f i x"
also have "\<dots> \<le> f (Suc i) x"
using \<open>incseq f\<close>[THEN incseq_SucD] unfolding le_fun_def by auto
finally show "a * u x \<le> f (Suc i) x" .
qed }
note B_mono = this
note B_u = sets.Int[OF u(1)[THEN simple_functionD(2)] B]
let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
proof -
fix i
have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
proof safe
fix x i assume x: "x \<in> space M"
show "x \<in> (\<Union>i. ?B' (u x) i)"
proof cases
assume "u x = 0" thus ?thesis using \<open>x \<in> space M\<close> f(3) by simp
next
assume "u x \<noteq> 0"
with \<open>a < 1\<close> u_range[OF \<open>x \<in> space M\<close>]
have "a * u x < 1 * u x"
by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
finally obtain i where "a * u x < f i x" unfolding SUP_def
by (auto simp add: less_SUP_iff)
hence "a * u x \<le> f i x" by auto
thus ?thesis using \<open>x \<in> space M\<close> by auto
qed
qed
then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
qed
have "integral\<^sup>S M u = (SUP i. integral\<^sup>S M (?uB i))"
unfolding simple_integral_indicator[OF B \<open>simple_function M u\<close>]
proof (subst SUP_ereal_setsum, safe)
fix x n assume "x \<in> space M"
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)"
using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
next
show "integral\<^sup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
using measure_conv u_range B_u unfolding simple_integral_def
by (auto intro!: setsum.cong SUP_ereal_mult_left [symmetric])
qed
moreover
have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
apply (subst SUP_ereal_mult_left [symmetric])
proof (safe intro!: SUP_mono bexI)
fix i
have "a * integral\<^sup>S M (?uB i) = (\<integral>\<^sup>Sx. a * ?uB i x \<partial>M)"
using B \<open>simple_function M u\<close> u_range
by (subst simple_integral_mult) (auto split: split_indicator)
also have "\<dots> \<le> integral\<^sup>N M (f i)"
proof -
have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B \<open>0 < a\<close> u(1) by auto
show ?thesis using f(3) * u_range \<open>0 < a\<close>
by (subst nn_integral_eq_simple_integral[symmetric])
(auto intro!: nn_integral_mono split: split_indicator)
qed
finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>N M (f i)"
by auto
next
fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B \<open>0 < a\<close> u(1) u_range
by (intro simple_integral_nonneg) (auto split: split_indicator)
qed (insert \<open>0 < a\<close>, auto)
ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
qed
lemma incseq_nn_integral:
assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>N M (f i))"
proof -
have "\<And>i x. f i x \<le> f (Suc i) x"
using assms by (auto dest!: incseq_SucD simp: le_fun_def)
then show ?thesis
by (auto intro!: incseq_SucI nn_integral_mono)
qed
lemma nn_integral_max_0: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>N M f"
by (simp add: le_fun_def nn_integral_def)
text \<open>Beppo-Levi monotone convergence theorem\<close>
lemma nn_integral_monotone_convergence_SUP:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M"
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))"
proof (rule antisym)
show "(SUP j. integral\<^sup>N M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
by (auto intro!: SUP_least SUP_upper nn_integral_mono)
next
have f': "incseq (\<lambda>i x. max 0 (f i x))"
using f by (auto simp: incseq_def le_fun_def not_le split: split_max)
(blast intro: order_trans less_imp_le)
have "(\<integral>\<^sup>+ x. max 0 (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. max 0 (f i x)) \<partial>M)"
unfolding sup_max[symmetric] Complete_Lattices.SUP_sup_distrib[symmetric] by simp
also have "\<dots> \<le> (SUP i. (\<integral>\<^sup>+ x. max 0 (f i x) \<partial>M))"
unfolding nn_integral_def_finite[of _ "\<lambda>x. SUP i. max 0 (f i x)"]
proof (safe intro!: SUP_least)
fix g assume g: "simple_function M g"
and *: "g \<le> max 0 \<circ> (\<lambda>x. SUP i. max 0 (f i x))" "range g \<subseteq> {0..<\<infinity>}"
then have "\<And>x. 0 \<le> (SUP i. max 0 (f i x))" and g': "g`space M \<subseteq> {0..<\<infinity>}"
using f by (auto intro!: SUP_upper2)
with * show "integral\<^sup>S M g \<le> (SUP j. \<integral>\<^sup>+x. max 0 (f j x) \<partial>M)"
by (intro nn_integral_SUP_approx[OF f' _ _ g _ g'])
(auto simp: le_fun_def max_def intro!: measurable_If f borel_measurable_le)
qed
finally show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>N M (f j))"
unfolding nn_integral_max_0 .
qed
lemma nn_integral_monotone_convergence_SUP_AE:
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"
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))"
proof -
from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
by (simp add: AE_all_countable)
from this[THEN AE_E] guess N . note N = this
let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
then have "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (SUP i. ?f i x) \<partial>M)"
by (auto intro!: nn_integral_cong_AE)
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
proof (rule nn_integral_monotone_convergence_SUP)
show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
{ fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
using f N(3) by (intro measurable_If_set) auto }
qed
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. f i x \<partial>M))"
using f_eq by (force intro!: arg_cong[where f="SUPREMUM UNIV"] nn_integral_cong_AE ext)
finally show ?thesis .
qed
lemma nn_integral_monotone_convergence_SUP_AE_incseq:
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"
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>N M (f i))"
using f[unfolded incseq_Suc_iff le_fun_def]
by (intro nn_integral_monotone_convergence_SUP_AE[OF _ borel])
auto
lemma nn_integral_monotone_convergence_simple:
assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
shows "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
using assms unfolding nn_integral_monotone_convergence_SUP[OF f(1)
f(3)[THEN borel_measurable_simple_function]]
by (auto intro!: nn_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPREMUM UNIV"] ext)
lemma nn_integral_cong_pos:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
shows "integral\<^sup>N M f = integral\<^sup>N M g"
proof -
have "integral\<^sup>N M (\<lambda>x. max 0 (f x)) = integral\<^sup>N M (\<lambda>x. max 0 (g x))"
proof (intro nn_integral_cong)
fix x assume "x \<in> space M"
from assms[OF this] show "max 0 (f x) = max 0 (g x)"
by (auto split: split_max)
qed
then show ?thesis by (simp add: nn_integral_max_0)
qed
lemma SUP_simple_integral_sequences:
assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
shows "(SUP i. integral\<^sup>S M (f i)) = (SUP i. integral\<^sup>S M (g i))"
(is "SUPREMUM _ ?F = SUPREMUM _ ?G")
proof -
have "(SUP i. integral\<^sup>S M (f i)) = (\<integral>\<^sup>+x. (SUP i. f i x) \<partial>M)"
using f by (rule nn_integral_monotone_convergence_simple)
also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
unfolding eq[THEN nn_integral_cong_AE] ..
also have "\<dots> = (SUP i. ?G i)"
using g by (rule nn_integral_monotone_convergence_simple[symmetric])
finally show ?thesis by simp
qed
lemma nn_integral_const[simp]:
"0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
by (subst nn_integral_eq_simple_integral) auto
lemma nn_integral_const_nonpos: "c \<le> 0 \<Longrightarrow> nn_integral M (\<lambda>x. c) = 0"
using nn_integral_max_0[of M "\<lambda>x. c"] by (simp add: max_def split: split_if_asm)
lemma nn_integral_linear:
assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
shows "(\<integral>\<^sup>+ x. a * f x + g x \<partial>M) = a * integral\<^sup>N M f + integral\<^sup>N M g"
(is "integral\<^sup>N M ?L = _")
proof -
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
note u = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
note v = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this
let ?L' = "\<lambda>i x. a * u i x + v i x"
have "?L \<in> borel_measurable M" using assms by auto
from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
note l = nn_integral_monotone_convergence_simple[OF this(2,5,1)] this
have inc: "incseq (\<lambda>i. a * integral\<^sup>S M (u i))" "incseq (\<lambda>i. integral\<^sup>S M (v i))"
using u v \<open>0 \<le> a\<close>
by (auto simp: incseq_Suc_iff le_fun_def
intro!: add_mono ereal_mult_left_mono simple_integral_mono)
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)"
using u v \<open>0 \<le> a\<close> by (auto simp: simple_integral_nonneg)
{ fix i from pos[of i] have "a * integral\<^sup>S M (u i) \<noteq> -\<infinity>" "integral\<^sup>S M (v i) \<noteq> -\<infinity>"
by (auto split: split_if_asm) }
note not_MInf = this
have l': "(SUP i. integral\<^sup>S M (l i)) = (SUP i. integral\<^sup>S M (?L' i))"
proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
using u v \<open>0 \<le> a\<close> unfolding incseq_Suc_iff le_fun_def
by (auto intro!: add_mono ereal_mult_left_mono)
{ fix x
{ fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using \<open>0 \<le> a\<close> u(6)[of i x] v(6)[of i x]
by auto }
then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
using \<open>0 \<le> a\<close> u(3) v(3) u(6)[of _ x] v(6)[of _ x]
by (subst SUP_ereal_mult_left [symmetric, OF _ u(6) \<open>0 \<le> a\<close>])
(auto intro!: SUP_ereal_add
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono) }
then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
unfolding l(5) using \<open>0 \<le> a\<close> u(5) v(5) l(5) f(2) g(2)
by (intro AE_I2) (auto split: split_max)
qed
also have "\<dots> = (SUP i. a * integral\<^sup>S M (u i) + integral\<^sup>S M (v i))"
using u(2, 6) v(2, 6) \<open>0 \<le> a\<close> by (auto intro!: arg_cong[where f="SUPREMUM UNIV"] ext)
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)"
unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
apply (subst SUP_ereal_mult_left [symmetric, OF _ pos(1) \<open>0 \<le> a\<close>])
apply simp
apply (subst SUP_ereal_add [symmetric, OF inc not_MInf])
.
then show ?thesis by (simp add: nn_integral_max_0)
qed
lemma nn_integral_cmult:
assumes f: "f \<in> borel_measurable M" "0 \<le> c"
shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>N M f"
proof -
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using \<open>0 \<le> c\<close>
by (auto split: split_max simp: ereal_zero_le_0_iff)
have "(\<integral>\<^sup>+ x. c * f x \<partial>M) = (\<integral>\<^sup>+ x. c * max 0 (f x) \<partial>M)"
by (simp add: nn_integral_max_0)
then show ?thesis
using nn_integral_linear[OF _ _ \<open>0 \<le> c\<close>, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
by (auto simp: nn_integral_max_0)
qed
lemma nn_integral_multc:
assumes "f \<in> borel_measurable M" "0 \<le> c"
shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>N M f * c"
unfolding mult.commute[of _ c] nn_integral_cmult[OF assms] by simp
lemma nn_integral_divide:
"\<lbrakk> 0 \<le> c; f \<in> borel_measurable M \<rbrakk> \<Longrightarrow> (\<integral>\<^sup>+ x. f x / c \<partial>M) = (\<integral>\<^sup>+ x. f x \<partial>M) / c"
by(simp add: divide_ereal_def nn_integral_multc inverse_ereal_ge0I)
lemma nn_integral_indicator[simp]:
"A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
by (subst nn_integral_eq_simple_integral)
(auto simp: simple_integral_indicator)
lemma nn_integral_cmult_indicator:
"0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
by (subst nn_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
lemma nn_integral_indicator':
assumes [measurable]: "A \<inter> space M \<in> sets M"
shows "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = emeasure M (A \<inter> space M)"
proof -
have "(\<integral>\<^sup>+ x. indicator A x \<partial>M) = (\<integral>\<^sup>+ x. indicator (A \<inter> space M) x \<partial>M)"
by (intro nn_integral_cong) (simp split: split_indicator)
also have "\<dots> = emeasure M (A \<inter> space M)"
by simp
finally show ?thesis .
qed
lemma nn_integral_add:
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
shows "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = integral\<^sup>N M f + integral\<^sup>N M g"
proof -
have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
using assms by (auto split: split_max)
have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
by (simp add: nn_integral_max_0)
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
unfolding ae[THEN nn_integral_cong_AE] ..
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (g x) \<partial>M)"
using nn_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
by auto
finally show ?thesis
by (simp add: nn_integral_max_0)
qed
lemma nn_integral_setsum:
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"
shows "(\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^sup>N M (f i))"
proof cases
assume f: "finite P"
from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
from f this assms(1) show ?thesis
proof induct
case (insert i P)
then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
"(\<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)"
by (auto intro!: setsum_nonneg)
from nn_integral_add[OF this]
show ?case using insert by auto
qed simp
qed simp
lemma nn_integral_bound_simple_function:
assumes bnd: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" "\<And>x. x \<in> space M \<Longrightarrow> f x < \<infinity>"
assumes f[measurable]: "simple_function M f"
assumes supp: "emeasure M {x\<in>space M. f x \<noteq> 0} < \<infinity>"
shows "nn_integral M f < \<infinity>"
proof cases
assume "space M = {}"
then have "nn_integral M f = (\<integral>\<^sup>+x. 0 \<partial>M)"
by (intro nn_integral_cong) auto
then show ?thesis by simp
next
assume "space M \<noteq> {}"
with simple_functionD(1)[OF f] bnd have bnd: "0 \<le> Max (f`space M) \<and> Max (f`space M) < \<infinity>"
by (subst Max_less_iff) (auto simp: Max_ge_iff)
have "nn_integral M f \<le> (\<integral>\<^sup>+x. Max (f`space M) * indicator {x\<in>space M. f x \<noteq> 0} x \<partial>M)"
proof (rule nn_integral_mono)
fix x assume "x \<in> space M"
with f show "f x \<le> Max (f ` space M) * indicator {x \<in> space M. f x \<noteq> 0} x"
by (auto split: split_indicator intro!: Max_ge simple_functionD)
qed
also have "\<dots> < \<infinity>"
using bnd supp by (subst nn_integral_cmult) auto
finally show ?thesis .
qed
lemma nn_integral_Markov_inequality:
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"
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)"
(is "(emeasure M) ?A \<le> _ * ?PI")
proof -
have "?A \<in> sets M"
using \<open>A \<in> sets M\<close> u by auto
hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
using nn_integral_indicator by simp
also have "\<dots> \<le> (\<integral>\<^sup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
by (auto intro!: nn_integral_mono_AE
simp: indicator_def ereal_zero_le_0_iff)
also have "\<dots> = c * (\<integral>\<^sup>+ x. u x * indicator A x \<partial>M)"
using assms
by (auto intro!: nn_integral_cmult simp: ereal_zero_le_0_iff)
finally show ?thesis .
qed
lemma nn_integral_noteq_infinite:
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
and "integral\<^sup>N M g \<noteq> \<infinity>"
shows "AE x in M. g x \<noteq> \<infinity>"
proof (rule ccontr)
assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
using c g by (auto simp add: AE_iff_null)
moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
also have "\<dots> \<le> (\<integral>\<^sup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
using g by (subst nn_integral_cmult_indicator) auto
also have "\<dots> \<le> integral\<^sup>N M g"
using assms by (auto intro!: nn_integral_mono_AE simp: indicator_def)
finally show False using \<open>integral\<^sup>N M g \<noteq> \<infinity>\<close> by auto
qed
lemma nn_integral_PInf:
assumes f: "f \<in> borel_measurable M"
and not_Inf: "integral\<^sup>N M f \<noteq> \<infinity>"
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
proof -
have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^sup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
using f by (subst nn_integral_cmult_indicator) (auto simp: measurable_sets)
also have "\<dots> \<le> integral\<^sup>N M (\<lambda>x. max 0 (f x))"
by (auto intro!: nn_integral_mono simp: indicator_def max_def)
finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>N M f"
by (simp add: nn_integral_max_0)
moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
by (rule emeasure_nonneg)
ultimately show ?thesis
using assms by (auto split: split_if_asm)
qed
lemma nn_integral_PInf_AE:
assumes "f \<in> borel_measurable M" "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
proof (rule AE_I)
show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
by (rule nn_integral_PInf[OF assms])
show "f -` {\<infinity>} \<inter> space M \<in> sets M"
using assms by (auto intro: borel_measurable_vimage)
qed auto
lemma simple_integral_PInf:
assumes "simple_function M f" "\<And>x. 0 \<le> f x"
and "integral\<^sup>S M f \<noteq> \<infinity>"
shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
proof (rule nn_integral_PInf)
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
show "integral\<^sup>N M f \<noteq> \<infinity>"
using assms by (simp add: nn_integral_eq_simple_integral)
qed
lemma nn_integral_diff:
assumes f: "f \<in> borel_measurable M"
and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
and fin: "integral\<^sup>N M g \<noteq> \<infinity>"
and mono: "AE x in M. g x \<le> f x"
shows "(\<integral>\<^sup>+ x. f x - g x \<partial>M) = integral\<^sup>N M f - integral\<^sup>N M g"
proof -
have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
using assms by (auto intro: ereal_diff_positive)
have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
{ fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
by (cases rule: ereal2_cases[of a b]) auto }
note * = this
then have "AE x in M. f x = f x - g x + g x"
using mono nn_integral_noteq_infinite[OF g fin] assms by auto
then have **: "integral\<^sup>N M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>N M g"
unfolding nn_integral_add[OF diff g, symmetric]
by (rule nn_integral_cong_AE)
show ?thesis unfolding **
using fin nn_integral_nonneg[of M g]
by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>N M g"]) auto
qed
lemma nn_integral_suminf:
assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
shows "(\<integral>\<^sup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>N M (f i))"
proof -
have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
using assms by (auto simp: AE_all_countable)
have "(\<Sum>i. integral\<^sup>N M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>N M (f i))"
using nn_integral_nonneg by (rule suminf_ereal_eq_SUP)
also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
unfolding nn_integral_setsum[OF f] ..
also have "\<dots> = \<integral>\<^sup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
by (intro nn_integral_monotone_convergence_SUP_AE[symmetric])
(elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
also have "\<dots> = \<integral>\<^sup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
by (intro nn_integral_cong_AE) (auto simp: suminf_ereal_eq_SUP)
finally show ?thesis by simp
qed
lemma nn_integral_mult_bounded_inf:
assumes f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
and c: "0 \<le> c" "c \<noteq> \<infinity>" and ae: "AE x in M. g x \<le> c * f x"
shows "(\<integral>\<^sup>+x. g x \<partial>M) < \<infinity>"
proof -
have "(\<integral>\<^sup>+x. g x \<partial>M) \<le> (\<integral>\<^sup>+x. c * f x \<partial>M)"
by (intro nn_integral_mono_AE ae)
also have "(\<integral>\<^sup>+x. c * f x \<partial>M) < \<infinity>"
using c f by (subst nn_integral_cmult) auto
finally show ?thesis .
qed
text \<open>Fatou's lemma: convergence theorem on limes inferior\<close>
lemma nn_integral_liminf:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
shows "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))"
proof -
have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
(SUP n. \<integral>\<^sup>+ x. (INF i:{n..}. u i x) \<partial>M)"
unfolding liminf_SUP_INF using pos u
by (intro nn_integral_monotone_convergence_SUP_AE)
(elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
also have "\<dots> \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))"
unfolding liminf_SUP_INF
by (auto intro!: SUP_mono exI INF_greatest nn_integral_mono INF_lower)
finally show ?thesis .
qed
lemma le_Limsup:
"F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. c \<le> g x) F \<Longrightarrow> c \<le> Limsup F g"
using Limsup_mono[of "\<lambda>_. c" g F] by (simp add: Limsup_const)
lemma Limsup_le:
"F \<noteq> bot \<Longrightarrow> eventually (\<lambda>x. f x \<le> c) F \<Longrightarrow> Limsup F f \<le> c"
using Limsup_mono[of f "\<lambda>_. c" F] by (simp add: Limsup_const)
lemma ereal_mono_minus_cancel:
fixes a b c :: ereal
shows "c - a \<le> c - b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> b \<le> a"
by (cases a b c rule: ereal3_cases) auto
lemma nn_integral_limsup:
fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes [measurable]: "\<And>i. u i \<in> borel_measurable M" "w \<in> borel_measurable M"
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>"
shows "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)"
proof -
have bnd: "AE x in M. \<forall>i. 0 \<le> u i x \<and> u i x \<le> w x"
using bounds by (auto simp: AE_all_countable)
from bounds[of 0] have w_nonneg: "AE x in M. 0 \<le> w x"
by auto
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)"
proof (intro nn_integral_diff[symmetric])
show "AE x in M. 0 \<le> limsup (\<lambda>n. u n x)"
using bnd by (auto intro!: le_Limsup)
show "AE x in M. limsup (\<lambda>n. u n x) \<le> w x"
using bnd by (auto intro!: Limsup_le)
then have "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) < \<infinity>"
by (intro nn_integral_mult_bounded_inf[OF _ w, of 1]) auto
then show "(\<integral>\<^sup>+x. limsup (\<lambda>n. u n x) \<partial>M) \<noteq> \<infinity>"
by simp
qed auto
also have "\<dots> = (\<integral>\<^sup>+x. liminf (\<lambda>n. w x - u n x) \<partial>M)"
using w_nonneg
by (intro nn_integral_cong_AE, eventually_elim)
(auto intro!: liminf_ereal_cminus[symmetric])
also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+x. w x - u n x \<partial>M)"
proof (rule nn_integral_liminf)
fix i show "AE x in M. 0 \<le> w x - u i x"
using bounds[of i] by eventually_elim (auto intro: ereal_diff_positive)
qed simp
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))"
proof (intro ext nn_integral_diff)
fix i have "(\<integral>\<^sup>+x. u i x \<partial>M) < \<infinity>"
using bounds by (intro nn_integral_mult_bounded_inf[OF _ w, of 1]) auto
then show "(\<integral>\<^sup>+x. u i x \<partial>M) \<noteq> \<infinity>" by simp
qed (insert bounds, auto)
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)"
using w by (intro liminf_ereal_cminus) auto
finally show ?thesis
by (rule ereal_mono_minus_cancel) (intro w nn_integral_nonneg)+
qed
lemma nn_integral_LIMSEQ:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>n x. 0 \<le> f n x"
and u: "\<And>x. (\<lambda>i. f i x) ----> u x"
shows "(\<lambda>n. integral\<^sup>N M (f n)) ----> integral\<^sup>N M u"
proof -
have "(\<lambda>n. integral\<^sup>N M (f n)) ----> (SUP n. integral\<^sup>N M (f n))"
using f by (intro LIMSEQ_SUP[of "\<lambda>n. integral\<^sup>N M (f n)"] incseq_nn_integral)
also have "(SUP n. integral\<^sup>N M (f n)) = integral\<^sup>N M (\<lambda>x. SUP n. f n x)"
using f by (intro nn_integral_monotone_convergence_SUP[symmetric])
also have "integral\<^sup>N M (\<lambda>x. SUP n. f n x) = integral\<^sup>N M (\<lambda>x. u x)"
using f by (subst SUP_Lim_ereal[OF _ u]) (auto simp: incseq_def le_fun_def)
finally show ?thesis .
qed
lemma nn_integral_dominated_convergence:
assumes [measurable]:
"\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
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"
and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
shows "(\<lambda>i. (\<integral>\<^sup>+x. u i x \<partial>M)) ----> (\<integral>\<^sup>+x. u' x \<partial>M)"
proof -
have "limsup (\<lambda>n. integral\<^sup>N M (u n)) \<le> (\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M)"
by (intro nn_integral_limsup[OF _ _ bound w]) auto
moreover have "(\<integral>\<^sup>+ x. limsup (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)"
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
moreover have "(\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M) = (\<integral>\<^sup>+ x. u' x \<partial>M)"
using u' by (intro nn_integral_cong_AE, eventually_elim) (metis tendsto_iff_Liminf_eq_Limsup sequentially_bot)
moreover have "(\<integral>\<^sup>+x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))"
by (intro nn_integral_liminf[OF _ bound(1)]) auto
moreover have "liminf (\<lambda>n. integral\<^sup>N M (u n)) \<le> limsup (\<lambda>n. integral\<^sup>N M (u n))"
by (intro Liminf_le_Limsup sequentially_bot)
ultimately show ?thesis
by (intro Liminf_eq_Limsup) auto
qed
lemma nn_integral_monotone_convergence_INF':
assumes f: "decseq f" and [measurable]: "\<And>i. f i \<in> borel_measurable M"
assumes "(\<integral>\<^sup>+ x. f 0 x \<partial>M) < \<infinity>" and nn: "\<And>x i. 0 \<le> f i x"
shows "(\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M) = (INF i. integral\<^sup>N M (f i))"
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>N M (f i)) ----> (INF i. integral\<^sup>N M (f i))"
using f by (intro LIMSEQ_INF) (auto intro!: nn_integral_mono simp: decseq_def le_fun_def)
show "(\<lambda>i. integral\<^sup>N M (f i)) ----> \<integral>\<^sup>+ x. (INF i. f i x) \<partial>M"
proof (rule nn_integral_dominated_convergence)
show "(\<integral>\<^sup>+ x. f 0 x \<partial>M) < \<infinity>" "\<And>i. f i \<in> borel_measurable M" "f 0 \<in> borel_measurable M"
by fact+
show "\<And>j. AE x in M. 0 \<le> f j x"
using nn by auto
show "\<And>j. AE x in M. f j x \<le> f 0 x"
using f by (auto simp: decseq_def le_fun_def)
show "AE x in M. (\<lambda>i. f i x) ----> (INF i. f i x)"
using f by (auto intro!: LIMSEQ_INF simp: decseq_def le_fun_def)
show "(\<lambda>x. INF i. f i x) \<in> borel_measurable M"
by auto
qed
qed
lemma nn_integral_monotone_convergence_INF:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
assumes f: "\<And>i j x. i \<le> j \<Longrightarrow> x \<in> space M \<Longrightarrow> f j x \<le> f i x"
and [measurable]: "\<And>i. f i \<in> borel_measurable M"
and fin: "(\<integral>\<^sup>+ x. f i x \<partial>M) < \<infinity>"
shows "(\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M) = (INF i. integral\<^sup>N M (f i))"
proof -
{ fix f :: "nat \<Rightarrow> ereal" and j assume "decseq f"
then have "(INF i. f i) = (INF i. f (i + j))"
apply (intro INF_eq)
apply (rule_tac x="i" in bexI)
apply (auto simp: decseq_def le_fun_def)
done }
note INF_shift = this
have dec: "decseq (\<lambda>j x. max 0 (restrict (f (j + i)) (space M) x))"
using f by (intro antimonoI le_funI max.mono) (auto simp: decseq_def le_fun_def)
have "(\<integral>\<^sup>+ x. max 0 (INF i. f i x) \<partial>M) = (\<integral>\<^sup>+ x. (INF i. max 0 (restrict (f i) (space M) x)) \<partial>M)"
by (intro nn_integral_cong)
(simp add: sup_ereal_def[symmetric] sup_INF del: sup_ereal_def)
also have "\<dots> = (\<integral>\<^sup>+ x. (INF j. max 0 (restrict (f (j + i)) (space M) x)) \<partial>M)"
using f by (intro nn_integral_cong INF_shift antimonoI le_funI max.mono)
(auto simp: decseq_def le_fun_def)
also have "\<dots> = (INF j. (\<integral>\<^sup>+ x. max 0 (restrict (f (j + i)) (space M) x) \<partial>M))"
proof (rule nn_integral_monotone_convergence_INF')
show "(\<lambda>x. max 0 (restrict (f (j + i)) (space M) x)) \<in> borel_measurable M" for j
by (subst measurable_cong[where g="\<lambda>x. max 0 (f (j + i) x)"]) measurable
show "(\<integral>\<^sup>+ x. max 0 (restrict (f (0 + i)) (space M) x) \<partial>M) < \<infinity>"
using fin by (simp add: nn_integral_max_0 cong: nn_integral_cong)
qed (intro max.cobounded1 dec f)+
also have "\<dots> = (INF j. (\<integral>\<^sup>+ x. max 0 (restrict (f j) (space M) x) \<partial>M))"
using f by (intro INF_shift[symmetric] nn_integral_mono antimonoI le_funI max.mono)
(auto simp: decseq_def le_fun_def)
finally show ?thesis unfolding nn_integral_max_0 by (simp cong: nn_integral_cong)
qed
lemma nn_integral_monotone_convergence_INF_decseq:
assumes f: "decseq f" and *: "\<And>i. f i \<in> borel_measurable M" "(\<integral>\<^sup>+ x. f i x \<partial>M) < \<infinity>"
shows "(\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M) = (INF i. integral\<^sup>N M (f i))"
using nn_integral_monotone_convergence_INF[of M f i, OF _ *] f by (auto simp: antimono_def le_fun_def)
lemma sup_continuous_nn_integral[order_continuous_intros]:
assumes f: "\<And>y. sup_continuous (f y)"
assumes [measurable]: "\<And>x. (\<lambda>y. f y x) \<in> borel_measurable M"
shows "sup_continuous (\<lambda>x. (\<integral>\<^sup>+y. f y x \<partial>M))"
unfolding sup_continuous_def
proof safe
fix C :: "nat \<Rightarrow> 'b" assume C: "incseq C"
with sup_continuous_mono[OF f] show "(\<integral>\<^sup>+ y. f y (SUPREMUM UNIV C) \<partial>M) = (SUP i. \<integral>\<^sup>+ y. f y (C i) \<partial>M)"
unfolding sup_continuousD[OF f C]
by (subst nn_integral_monotone_convergence_SUP) (auto simp: mono_def le_fun_def)
qed
lemma inf_continuous_nn_integral[order_continuous_intros]:
assumes f: "\<And>y. inf_continuous (f y)"
assumes [measurable]: "\<And>x. (\<lambda>y. f y x) \<in> borel_measurable M"
assumes bnd: "\<And>x. (\<integral>\<^sup>+ y. f y x \<partial>M) \<noteq> \<infinity>"
shows "inf_continuous (\<lambda>x. (\<integral>\<^sup>+y. f y x \<partial>M))"
unfolding inf_continuous_def
proof safe
fix C :: "nat \<Rightarrow> 'b" assume C: "decseq C"
then show "(\<integral>\<^sup>+ y. f y (INFIMUM UNIV C) \<partial>M) = (INF i. \<integral>\<^sup>+ y. f y (C i) \<partial>M)"
using inf_continuous_mono[OF f]
by (auto simp add: inf_continuousD[OF f C] fun_eq_iff antimono_def mono_def le_fun_def bnd
intro!: nn_integral_monotone_convergence_INF)
qed
lemma nn_integral_null_set:
assumes "N \<in> null_sets M" shows "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = 0"
proof -
have "(\<integral>\<^sup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
proof (intro nn_integral_cong_AE AE_I)
show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
by (auto simp: indicator_def)
show "(emeasure M) N = 0" "N \<in> sets M"
using assms by auto
qed
then show ?thesis by simp
qed
lemma nn_integral_0_iff:
assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
(is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
proof -
have u_eq: "(\<integral>\<^sup>+ x. u x * indicator ?A x \<partial>M) = integral\<^sup>N M u"
by (auto intro!: nn_integral_cong simp: indicator_def)
show ?thesis
proof
assume "(emeasure M) ?A = 0"
with nn_integral_null_set[of ?A M u] u
show "integral\<^sup>N M u = 0" by (simp add: u_eq null_sets_def)
next
{ fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
note gt_1 = this
assume *: "integral\<^sup>N M u = 0"
let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
proof -
{ fix n :: nat
from nn_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
thus ?thesis by simp
qed
also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
proof (safe intro!: SUP_emeasure_incseq)
fix n show "?M n \<inter> ?A \<in> sets M"
using u by (auto intro!: sets.Int)
next
show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
proof (safe intro!: incseq_SucI)
fix n :: nat and x
assume *: "1 \<le> real n * u x"
also from gt_1[OF *] have "real n * u x \<le> real (Suc n) * u x"
using \<open>0 \<le> u x\<close> by (auto intro!: ereal_mult_right_mono)
finally show "1 \<le> real (Suc n) * u x" by auto
qed
qed
also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
proof (cases "u x")
case (real r) with \<open>0 < u x\<close> have "0 < r" by auto
obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using \<open>0 < r\<close> by auto
hence "1 \<le> real j * r" using real \<open>0 < r\<close> by auto
thus ?thesis using \<open>0 < r\<close> real by (auto simp: one_ereal_def)
qed (insert \<open>0 < u x\<close>, auto)
qed auto
finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
moreover
from pos have "AE x in M. \<not> (u x < 0)" by auto
then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
using AE_iff_null[of M] u by auto
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}"
using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
ultimately show "(emeasure M) ?A = 0" by simp
qed
qed
lemma nn_integral_0_iff_AE:
assumes u: "u \<in> borel_measurable M"
shows "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
proof -
have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
using u by auto
from nn_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
have "integral\<^sup>N M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
unfolding nn_integral_max_0
using AE_iff_null[OF sets] u by auto
also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
finally show ?thesis .
qed
lemma AE_iff_nn_integral:
"{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"
by (subst nn_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
lemma nn_integral_less:
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
assumes f: "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) \<noteq> \<infinity>"
assumes ord: "AE x in M. f x \<le> g x" "\<not> (AE x in M. g x \<le> f x)"
shows "(\<integral>\<^sup>+x. f x \<partial>M) < (\<integral>\<^sup>+x. g x \<partial>M)"
proof -
have "0 < (\<integral>\<^sup>+x. g x - f x \<partial>M)"
proof (intro order_le_neq_trans nn_integral_nonneg notI)
assume "0 = (\<integral>\<^sup>+x. g x - f x \<partial>M)"
then have "AE x in M. g x - f x \<le> 0"
using nn_integral_0_iff_AE[of "\<lambda>x. g x - f x" M] by simp
with f(1) ord(1) have "AE x in M. g x \<le> f x"
by eventually_elim (auto simp: ereal_minus_le_iff)
with ord show False
by simp
qed
also have "\<dots> = (\<integral>\<^sup>+x. g x \<partial>M) - (\<integral>\<^sup>+x. f x \<partial>M)"
by (subst nn_integral_diff) (auto simp: f ord)
finally show ?thesis
by (simp add: ereal_less_minus_iff f nn_integral_nonneg)
qed
lemma nn_integral_const_If:
"(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
by (auto intro!: nn_integral_0_iff_AE[THEN iffD2])
lemma nn_integral_subalgebra:
assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
shows "integral\<^sup>N N f = integral\<^sup>N M f"
proof -
have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
using N by (auto simp: measurable_def)
have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
using N by (auto simp add: eventually_ae_filter null_sets_def)
have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
using N by auto
from f show ?thesis
apply induct
apply (simp_all add: nn_integral_add nn_integral_cmult nn_integral_monotone_convergence_SUP N)
apply (auto intro!: nn_integral_cong cong: nn_integral_cong simp: N(2)[symmetric])
done
qed
lemma nn_integral_nat_function:
fixes f :: "'a \<Rightarrow> nat"
assumes "f \<in> measurable M (count_space UNIV)"
shows "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<Sum>t. emeasure M {x\<in>space M. t < f x})"
proof -
def F \<equiv> "\<lambda>i. {x\<in>space M. i < f x}"
with assms have [measurable]: "\<And>i. F i \<in> sets M"
by auto
{ fix x assume "x \<in> space M"
have "(\<lambda>i. if i < f x then 1 else 0) sums (of_nat (f x)::real)"
using sums_If_finite[of "\<lambda>i. i < f x" "\<lambda>_. 1::real"] by simp
then have "(\<lambda>i. ereal(if i < f x then 1 else 0)) sums (ereal(of_nat(f x)))"
unfolding sums_ereal .
moreover have "\<And>i. ereal (if i < f x then 1 else 0) = indicator (F i) x"
using \<open>x \<in> space M\<close> by (simp add: one_ereal_def F_def)
ultimately have "ereal(of_nat(f x)) = (\<Sum>i. indicator (F i) x)"
by (simp add: sums_iff) }
then have "(\<integral>\<^sup>+x. ereal (of_nat (f x)) \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>i. indicator (F i) x) \<partial>M)"
by (simp cong: nn_integral_cong)
also have "\<dots> = (\<Sum>i. emeasure M (F i))"
by (simp add: nn_integral_suminf)
finally show ?thesis
by (simp add: F_def)
qed
lemma nn_integral_lfp:
assumes sets[simp]: "\<And>s. sets (M s) = sets N"
assumes f: "sup_continuous f"
assumes g: "sup_continuous g"
assumes nonneg: "\<And>F s. 0 \<le> g F s"
assumes meas: "\<And>F. F \<in> borel_measurable N \<Longrightarrow> f F \<in> borel_measurable N"
assumes step: "\<And>F s. F \<in> borel_measurable N \<Longrightarrow> integral\<^sup>N (M s) (f F) = g (\<lambda>s. integral\<^sup>N (M s) F) s"
shows "(\<integral>\<^sup>+\<omega>. lfp f \<omega> \<partial>M s) = lfp g s"
proof (subst lfp_transfer_bounded[where \<alpha>="\<lambda>F s. \<integral>\<^sup>+x. F x \<partial>M s" and g=g and f=f and P="\<lambda>f. f \<in> borel_measurable N", symmetric])
fix C :: "nat \<Rightarrow> 'b \<Rightarrow> ereal" assume "incseq C" "\<And>i. C i \<in> borel_measurable N"
then show "(\<lambda>s. \<integral>\<^sup>+x. (SUP i. C i) x \<partial>M s) = (SUP i. (\<lambda>s. \<integral>\<^sup>+x. C i x \<partial>M s))"
unfolding SUP_apply[abs_def]
by (subst nn_integral_monotone_convergence_SUP)
(auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets)
next
show "\<And>x. (\<lambda>s. integral\<^sup>N (M s) bot) \<le> g x"
by (subst nn_integral_max_0[symmetric])
(simp add: sup_ereal_def[symmetric] le_fun_def nonneg del: sup_ereal_def)
qed (auto simp add: step nonneg le_fun_def SUP_apply[abs_def] bot_fun_def intro!: meas f g)
lemma nn_integral_gfp:
assumes sets[simp]: "\<And>s. sets (M s) = sets N"
assumes f: "inf_continuous f" and g: "inf_continuous g"
assumes meas: "\<And>F. F \<in> borel_measurable N \<Longrightarrow> f F \<in> borel_measurable N"
assumes bound: "\<And>F s. F \<in> borel_measurable N \<Longrightarrow> (\<integral>\<^sup>+x. f F x \<partial>M s) < \<infinity>"
assumes non_zero: "\<And>s. emeasure (M s) (space (M s)) \<noteq> 0"
assumes step: "\<And>F s. F \<in> borel_measurable N \<Longrightarrow> integral\<^sup>N (M s) (f F) = g (\<lambda>s. integral\<^sup>N (M s) F) s"
shows "(\<integral>\<^sup>+\<omega>. gfp f \<omega> \<partial>M s) = gfp g s"
proof (subst gfp_transfer_bounded[where \<alpha>="\<lambda>F s. \<integral>\<^sup>+x. F x \<partial>M s" and g=g and f=f
and P="\<lambda>F. F \<in> borel_measurable N \<and> (\<forall>s. (\<integral>\<^sup>+x. F x \<partial>M s) < \<infinity>)", symmetric])
fix C :: "nat \<Rightarrow> 'b \<Rightarrow> ereal" assume "decseq C" "\<And>i. C i \<in> borel_measurable N \<and> (\<forall>s. integral\<^sup>N (M s) (C i) < \<infinity>)"
then show "(\<lambda>s. \<integral>\<^sup>+x. (INF i. C i) x \<partial>M s) = (INF i. (\<lambda>s. \<integral>\<^sup>+x. C i x \<partial>M s))"
unfolding INF_apply[abs_def]
by (subst nn_integral_monotone_convergence_INF_decseq)
(auto simp: mono_def fun_eq_iff intro!: arg_cong2[where f=emeasure] cong: measurable_cong_sets)
next
show "\<And>x. g x \<le> (\<lambda>s. integral\<^sup>N (M s) (f top))"
by (subst step)
(auto simp add: top_fun_def top_ereal_def less_le emeasure_nonneg non_zero le_fun_def
cong del: if_cong intro!: monoD[OF inf_continuous_mono[OF g], THEN le_funD])
next
fix C assume "\<And>i::nat. C i \<in> borel_measurable N \<and> (\<forall>s. integral\<^sup>N (M s) (C i) < \<infinity>)" "decseq C"
with bound show "INFIMUM UNIV C \<in> borel_measurable N \<and> (\<forall>s. integral\<^sup>N (M s) (INFIMUM UNIV C) < \<infinity>)"
unfolding INF_apply[abs_def]
by (subst nn_integral_monotone_convergence_INF_decseq)
(auto cong: measurable_cong_sets intro!: borel_measurable_INF
simp: INF_less_iff simp del: ereal_infty_less(1))
next
show "\<And>x. x \<in> borel_measurable N \<and> (\<forall>s. integral\<^sup>N (M s) x < \<infinity>) \<Longrightarrow>
(\<lambda>s. integral\<^sup>N (M s) (f x)) = g (\<lambda>s. integral\<^sup>N (M s) x)"
by (subst step) auto
qed (insert bound, auto simp add: le_fun_def INF_apply[abs_def] top_fun_def intro!: meas f g)
subsection \<open>Integral under concrete measures\<close>
lemma nn_integral_empty:
assumes "space M = {}"
shows "nn_integral M f = 0"
proof -
have "(\<integral>\<^sup>+ x. f x \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
by(rule nn_integral_cong)(simp add: assms)
thus ?thesis by simp
qed
subsubsection \<open>Distributions\<close>
lemma nn_integral_distr':
assumes T: "T \<in> measurable M M'"
and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
shows "integral\<^sup>N (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
using f
proof induct
case (cong f g)
with T show ?case
apply (subst nn_integral_cong[of _ f g])
apply simp
apply (subst nn_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
apply (simp add: measurable_def Pi_iff)
apply simp
done
next
case (set A)
then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
by (auto simp: indicator_def)
from set T show ?case
by (subst nn_integral_cong[OF eq])
(auto simp add: emeasure_distr intro!: nn_integral_indicator[symmetric] measurable_sets)
qed (simp_all add: measurable_compose[OF T] T nn_integral_cmult nn_integral_add
nn_integral_monotone_convergence_SUP le_fun_def incseq_def)
lemma nn_integral_distr:
"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)"
by (subst (1 2) nn_integral_max_0[symmetric])
(simp add: nn_integral_distr')
subsubsection \<open>Counting space\<close>
lemma simple_function_count_space[simp]:
"simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
unfolding simple_function_def by simp
lemma nn_integral_count_space:
assumes A: "finite {a\<in>A. 0 < f a}"
shows "integral\<^sup>N (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
proof -
have *: "(\<integral>\<^sup>+x. max 0 (f x) \<partial>count_space A) =
(\<integral>\<^sup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
by (auto intro!: nn_integral_cong
simp add: indicator_def if_distrib setsum.If_cases[OF A] max_def le_less)
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^sup>+ x. f a * indicator {a} x \<partial>count_space A)"
by (subst nn_integral_setsum)
(simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
by (auto intro!: setsum.cong simp: nn_integral_cmult_indicator one_ereal_def[symmetric])
finally show ?thesis by (simp add: nn_integral_max_0)
qed
lemma nn_integral_count_space_finite:
"finite A \<Longrightarrow> (\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
by (subst nn_integral_max_0[symmetric])
(auto intro!: setsum.mono_neutral_left simp: nn_integral_count_space less_le)
lemma nn_integral_count_space':
assumes "finite A" "\<And>x. x \<in> B \<Longrightarrow> x \<notin> A \<Longrightarrow> f x = 0" "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" "A \<subseteq> B"
shows "(\<integral>\<^sup>+x. f x \<partial>count_space B) = (\<Sum>x\<in>A. f x)"
proof -
have "(\<integral>\<^sup>+x. f x \<partial>count_space B) = (\<Sum>a | a \<in> B \<and> 0 < f a. f a)"
using assms(2,3)
by (intro nn_integral_count_space finite_subset[OF _ \<open>finite A\<close>]) (auto simp: less_le)
also have "\<dots> = (\<Sum>a\<in>A. f a)"
using assms by (intro setsum.mono_neutral_cong_left) (auto simp: less_le)
finally show ?thesis .
qed
lemma nn_integral_bij_count_space:
assumes g: "bij_betw g A B"
shows "(\<integral>\<^sup>+x. f (g x) \<partial>count_space A) = (\<integral>\<^sup>+x. f x \<partial>count_space B)"
using g[THEN bij_betw_imp_funcset]
by (subst distr_bij_count_space[OF g, symmetric])
(auto intro!: nn_integral_distr[symmetric])
lemma nn_integral_indicator_finite:
fixes f :: "'a \<Rightarrow> ereal"
assumes f: "finite A" and nn: "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" and [measurable]: "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"
shows "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<Sum>x\<in>A. f x * emeasure M {x})"
proof -
from f have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. (\<Sum>a\<in>A. f a * indicator {a} x) \<partial>M)"
by (intro nn_integral_cong) (auto simp: indicator_def if_distrib[where f="\<lambda>a. x * a" for x] setsum.If_cases)
also have "\<dots> = (\<Sum>a\<in>A. f a * emeasure M {a})"
using nn by (subst nn_integral_setsum) (auto simp: nn_integral_cmult_indicator)
finally show ?thesis .
qed
lemma nn_integral_count_space_nat:
fixes f :: "nat \<Rightarrow> ereal"
assumes nonneg: "\<And>i. 0 \<le> f i"
shows "(\<integral>\<^sup>+i. f i \<partial>count_space UNIV) = (\<Sum>i. f i)"
proof -
have "(\<integral>\<^sup>+i. f i \<partial>count_space UNIV) =
(\<integral>\<^sup>+i. (\<Sum>j. f j * indicator {j} i) \<partial>count_space UNIV)"
proof (intro nn_integral_cong)
fix i
have "f i = (\<Sum>j\<in>{i}. f j * indicator {j} i)"
by simp
also have "\<dots> = (\<Sum>j. f j * indicator {j} i)"
by (rule suminf_finite[symmetric]) auto
finally show "f i = (\<Sum>j. f j * indicator {j} i)" .
qed
also have "\<dots> = (\<Sum>j. (\<integral>\<^sup>+i. f j * indicator {j} i \<partial>count_space UNIV))"
by (rule nn_integral_suminf) (auto simp: nonneg)
also have "\<dots> = (\<Sum>j. f j)"
by (simp add: nonneg nn_integral_cmult_indicator one_ereal_def[symmetric])
finally show ?thesis .
qed
lemma nn_integral_count_space_nn_integral:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> ereal"
assumes "countable I" and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M"
shows "(\<integral>\<^sup>+x. \<integral>\<^sup>+i. f i x \<partial>count_space I \<partial>M) = (\<integral>\<^sup>+i. \<integral>\<^sup>+x. f i x \<partial>M \<partial>count_space I)"
proof cases
assume "finite I" then show ?thesis
by (simp add: nn_integral_count_space_finite nn_integral_nonneg max.absorb2 nn_integral_setsum
nn_integral_max_0)
next
assume "infinite I"
then have [simp]: "I \<noteq> {}"
by auto
note * = bij_betw_from_nat_into[OF \<open>countable I\<close> \<open>infinite I\<close>]
have **: "\<And>f. (\<And>i. 0 \<le> f i) \<Longrightarrow> (\<integral>\<^sup>+i. f i \<partial>count_space I) = (\<Sum>n. f (from_nat_into I n))"
by (simp add: nn_integral_bij_count_space[symmetric, OF *] nn_integral_count_space_nat)
show ?thesis
apply (subst (2) nn_integral_max_0[symmetric])
apply (simp add: ** nn_integral_nonneg nn_integral_suminf from_nat_into)
apply (simp add: nn_integral_max_0)
done
qed
lemma emeasure_UN_countable:
assumes sets[measurable]: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M" and I[simp]: "countable I"
assumes disj: "disjoint_family_on X I"
shows "emeasure M (UNION I X) = (\<integral>\<^sup>+i. emeasure M (X i) \<partial>count_space I)"
proof -
have eq: "\<And>x. indicator (UNION I X) x = \<integral>\<^sup>+ i. indicator (X i) x \<partial>count_space I"
proof cases
fix x assume x: "x \<in> UNION I X"
then obtain j where j: "x \<in> X j" "j \<in> I"
by auto
with disj have "\<And>i. i \<in> I \<Longrightarrow> indicator (X i) x = (indicator {j} i::ereal)"
by (auto simp: disjoint_family_on_def split: split_indicator)
with x j show "?thesis x"
by (simp cong: nn_integral_cong_simp)
qed (auto simp: nn_integral_0_iff_AE)
note sets.countable_UN'[unfolded subset_eq, measurable]
have "emeasure M (UNION I X) = (\<integral>\<^sup>+x. indicator (UNION I X) x \<partial>M)"
by simp
also have "\<dots> = (\<integral>\<^sup>+i. \<integral>\<^sup>+x. indicator (X i) x \<partial>M \<partial>count_space I)"
by (simp add: eq nn_integral_count_space_nn_integral)
finally show ?thesis
by (simp cong: nn_integral_cong_simp)
qed
lemma emeasure_countable_singleton:
assumes sets: "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" and X: "countable X"
shows "emeasure M X = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space X)"
proof -
have "emeasure M (\<Union>i\<in>X. {i}) = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space X)"
using assms by (intro emeasure_UN_countable) (auto simp: disjoint_family_on_def)
also have "(\<Union>i\<in>X. {i}) = X" by auto
finally show ?thesis .
qed
lemma measure_eqI_countable:
assumes [simp]: "sets M = Pow A" "sets N = Pow A" and A: "countable A"
assumes eq: "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}"
shows "M = N"
proof (rule measure_eqI)
fix X assume "X \<in> sets M"
then have X: "X \<subseteq> A" by auto
moreover with A have "countable X" by (auto dest: countable_subset)
ultimately have
"emeasure M X = (\<integral>\<^sup>+a. emeasure M {a} \<partial>count_space X)"
"emeasure N X = (\<integral>\<^sup>+a. emeasure N {a} \<partial>count_space X)"
by (auto intro!: emeasure_countable_singleton)
moreover have "(\<integral>\<^sup>+a. emeasure M {a} \<partial>count_space X) = (\<integral>\<^sup>+a. emeasure N {a} \<partial>count_space X)"
using X by (intro nn_integral_cong eq) auto
ultimately show "emeasure M X = emeasure N X"
by simp
qed simp
lemma measure_eqI_countable_AE:
assumes [simp]: "sets M = UNIV" "sets N = UNIV"
assumes ae: "AE x in M. x \<in> \<Omega>" "AE x in N. x \<in> \<Omega>" and [simp]: "countable \<Omega>"
assumes eq: "\<And>x. x \<in> \<Omega> \<Longrightarrow> emeasure M {x} = emeasure N {x}"
shows "M = N"
proof (rule measure_eqI)
fix A
have "emeasure N A = emeasure N {x\<in>\<Omega>. x \<in> A}"
using ae by (intro emeasure_eq_AE) auto
also have "\<dots> = (\<integral>\<^sup>+x. emeasure N {x} \<partial>count_space {x\<in>\<Omega>. x \<in> A})"
by (intro emeasure_countable_singleton) auto
also have "\<dots> = (\<integral>\<^sup>+x. emeasure M {x} \<partial>count_space {x\<in>\<Omega>. x \<in> A})"
by (intro nn_integral_cong eq[symmetric]) auto
also have "\<dots> = emeasure M {x\<in>\<Omega>. x \<in> A}"
by (intro emeasure_countable_singleton[symmetric]) auto
also have "\<dots> = emeasure M A"
using ae by (intro emeasure_eq_AE) auto
finally show "emeasure M A = emeasure N A" ..
qed simp
lemma nn_integral_monotone_convergence_SUP_nat':
fixes f :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
assumes chain: "Complete_Partial_Order.chain op \<le> (f ` Y)"
and nonempty: "Y \<noteq> {}"
and nonneg: "\<And>i n. i \<in> Y \<Longrightarrow> f i n \<ge> 0"
shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space UNIV) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space UNIV))"
(is "?lhs = ?rhs" is "integral\<^sup>N ?M _ = _")
proof (rule order_class.order.antisym)
show "?rhs \<le> ?lhs"
by (auto intro!: SUP_least SUP_upper nn_integral_mono)
next
have "\<And>x. \<exists>g. incseq g \<and> range g \<subseteq> (\<lambda>i. f i x) ` Y \<and> (SUP i:Y. f i x) = (SUP i. g i)"
unfolding Sup_class.SUP_def by(rule Sup_countable_SUP[unfolded Sup_class.SUP_def])(simp add: nonempty)
then obtain g where incseq: "\<And>x. incseq (g x)"
and range: "\<And>x. range (g x) \<subseteq> (\<lambda>i. f i x) ` Y"
and sup: "\<And>x. (SUP i:Y. f i x) = (SUP i. g x i)" by moura
from incseq have incseq': "incseq (\<lambda>i x. g x i)"
by(blast intro: incseq_SucI le_funI dest: incseq_SucD)
have "?lhs = \<integral>\<^sup>+ x. (SUP i. g x i) \<partial>?M" by(simp add: sup)
also have "\<dots> = (SUP i. \<integral>\<^sup>+ x. g x i \<partial>?M)" using incseq'
by(rule nn_integral_monotone_convergence_SUP) simp
also have "\<dots> \<le> (SUP i:Y. \<integral>\<^sup>+ x. f i x \<partial>?M)"
proof(rule SUP_least)
fix n
have "\<And>x. \<exists>i. g x n = f i x \<and> i \<in> Y" using range by blast
then obtain I where I: "\<And>x. g x n = f (I x) x" "\<And>x. I x \<in> Y" by moura
{ fix x
from range[of x] obtain i where "i \<in> Y" "g x n = f i x" by auto
hence "g x n \<ge> 0" using nonneg[of i x] by simp }
note nonneg_g = this
then have "(\<integral>\<^sup>+ x. g x n \<partial>count_space UNIV) = (\<Sum>x. g x n)"
by(rule nn_integral_count_space_nat)
also have "\<dots> = (SUP m. \<Sum>x<m. g x n)" using nonneg_g
by(rule suminf_ereal_eq_SUP)
also have "\<dots> \<le> (SUP i:Y. \<integral>\<^sup>+ x. f i x \<partial>?M)"
proof(rule SUP_mono)
fix m
show "\<exists>m'\<in>Y. (\<Sum>x<m. g x n) \<le> (\<integral>\<^sup>+ x. f m' x \<partial>?M)"
proof(cases "m > 0")
case False
thus ?thesis using nonempty by(auto simp add: nn_integral_nonneg)
next
case True
let ?Y = "I ` {..<m}"
have "f ` ?Y \<subseteq> f ` Y" using I by auto
with chain have chain': "Complete_Partial_Order.chain op \<le> (f ` ?Y)" by(rule chain_subset)
hence "Sup (f ` ?Y) \<in> f ` ?Y"
by(rule ccpo_class.in_chain_finite)(auto simp add: True lessThan_empty_iff)
then obtain m' where "m' < m" and m': "(SUP i:?Y. f i) = f (I m')" by auto
have "I m' \<in> Y" using I by blast
have "(\<Sum>x<m. g x n) \<le> (\<Sum>x<m. f (I m') x)"
proof(rule setsum_mono)
fix x
assume "x \<in> {..<m}"
hence "x < m" by simp
have "g x n = f (I x) x" by(simp add: I)
also have "\<dots> \<le> (SUP i:?Y. f i) x" unfolding SUP_def Sup_fun_def image_image
using \<open>x \<in> {..<m}\<close> by(rule Sup_upper[OF imageI])
also have "\<dots> = f (I m') x" unfolding m' by simp
finally show "g x n \<le> f (I m') x" .
qed
also have "\<dots> \<le> (SUP m. (\<Sum>x<m. f (I m') x))"
by(rule SUP_upper) simp
also have "\<dots> = (\<Sum>x. f (I m') x)"
by(rule suminf_ereal_eq_SUP[symmetric])(simp add: nonneg \<open>I m' \<in> Y\<close>)
also have "\<dots> = (\<integral>\<^sup>+ x. f (I m') x \<partial>?M)"
by(rule nn_integral_count_space_nat[symmetric])(simp add: nonneg \<open>I m' \<in> Y\<close>)
finally show ?thesis using \<open>I m' \<in> Y\<close> by blast
qed
qed
finally show "(\<integral>\<^sup>+ x. g x n \<partial>count_space UNIV) \<le> \<dots>" .
qed
finally show "?lhs \<le> ?rhs" .
qed
lemma nn_integral_monotone_convergence_SUP_nat:
fixes f :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
assumes nonempty: "Y \<noteq> {}"
and chain: "Complete_Partial_Order.chain op \<le> (f ` Y)"
shows "(\<integral>\<^sup>+ x. (SUP i:Y. f i x) \<partial>count_space UNIV) = (SUP i:Y. (\<integral>\<^sup>+ x. f i x \<partial>count_space UNIV))"
(is "?lhs = ?rhs")
proof -
let ?f = "\<lambda>i x. max 0 (f i x)"
have chain': "Complete_Partial_Order.chain op \<le> (?f ` Y)"
proof(rule chainI)
fix g h
assume "g \<in> ?f ` Y" "h \<in> ?f ` Y"
then obtain g' h' where gh: "g' \<in> Y" "h' \<in> Y" "g = ?f g'" "h = ?f h'" by blast
hence "f g' \<in> f ` Y" "f h' \<in> f ` Y" by blast+
with chain have "f g' \<le> f h' \<or> f h' \<le> f g'" by(rule chainD)
thus "g \<le> h \<or> h \<le> g"
proof
assume "f g' \<le> f h'"
hence "g \<le> h" using gh order_trans by(auto simp add: le_fun_def max_def)
thus ?thesis ..
next
assume "f h' \<le> f g'"
hence "h \<le> g" using gh order_trans by(auto simp add: le_fun_def max_def)
thus ?thesis ..
qed
qed
have "?lhs = (\<integral>\<^sup>+ x. max 0 (SUP i:Y. f i x) \<partial>count_space UNIV)"
by(simp add: nn_integral_max_0)
also have "\<dots> = (\<integral>\<^sup>+ x. (SUP i:Y. ?f i x) \<partial>count_space UNIV)"
proof(rule nn_integral_cong)
fix x
have "max 0 (SUP i:Y. f i x) \<le> (SUP i:Y. ?f i x)"
proof(cases "0 \<le> (SUP i:Y. f i x)")
case True
have "(SUP i:Y. f i x) \<le> (SUP i:Y. ?f i x)" by(rule SUP_mono)(auto intro: rev_bexI)
with True show ?thesis by simp
next
case False
have "0 \<le> (SUP i:Y. ?f i x)" using nonempty by(auto intro: SUP_upper2)
thus ?thesis using False by simp
qed
moreover have "\<dots> \<le> max 0 (SUP i:Y. f i x)"
proof(cases "(SUP i:Y. f i x) \<ge> 0")
case True
show ?thesis
by(rule SUP_least)(auto simp add: True max_def intro: SUP_upper)
next
case False
hence "(SUP i:Y. f i x) \<le> 0" by simp
hence less: "\<forall>i\<in>Y. f i x \<le> 0" by(simp add: SUP_le_iff)
show ?thesis by(rule SUP_least)(auto simp add: max_def less intro: SUP_upper)
qed
ultimately show "\<dots> = (SUP i:Y. ?f i x)" by(rule order.antisym)
qed
also have "\<dots> = (SUP i:Y. (\<integral>\<^sup>+ x. ?f i x \<partial>count_space UNIV))"
using chain' nonempty by(rule nn_integral_monotone_convergence_SUP_nat') simp
also have "\<dots> = ?rhs" by(simp add: nn_integral_max_0)
finally show ?thesis .
qed
subsubsection \<open>Measures with Restricted Space\<close>
lemma simple_function_iff_borel_measurable:
fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
shows "simple_function M f \<longleftrightarrow> finite (f ` space M) \<and> f \<in> borel_measurable M"
by (metis borel_measurable_simple_function simple_functionD(1) simple_function_borel_measurable)
lemma simple_function_restrict_space_ereal:
fixes f :: "'a \<Rightarrow> ereal"
assumes "\<Omega> \<inter> space M \<in> sets M"
shows "simple_function (restrict_space M \<Omega>) f \<longleftrightarrow> simple_function M (\<lambda>x. f x * indicator \<Omega> x)"
proof -
{ assume "finite (f ` space (restrict_space M \<Omega>))"
then have "finite (f ` space (restrict_space M \<Omega>) \<union> {0})" by simp
then have "finite ((\<lambda>x. f x * indicator \<Omega> x) ` space M)"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
moreover
{ assume "finite ((\<lambda>x. f x * indicator \<Omega> x) ` space M)"
then have "finite (f ` space (restrict_space M \<Omega>))"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
ultimately show ?thesis
unfolding simple_function_iff_borel_measurable
borel_measurable_restrict_space_iff_ereal[OF assms]
by auto
qed
lemma simple_function_restrict_space:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "\<Omega> \<inter> space M \<in> sets M"
shows "simple_function (restrict_space M \<Omega>) f \<longleftrightarrow> simple_function M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
proof -
{ assume "finite (f ` space (restrict_space M \<Omega>))"
then have "finite (f ` space (restrict_space M \<Omega>) \<union> {0})" by simp
then have "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x) ` space M)"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
moreover
{ assume "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x) ` space M)"
then have "finite (f ` space (restrict_space M \<Omega>))"
by (rule rev_finite_subset) (auto split: split_indicator simp: space_restrict_space) }
ultimately show ?thesis
unfolding simple_function_iff_borel_measurable
borel_measurable_restrict_space_iff[OF assms]
by auto
qed
lemma split_indicator_asm: "P (indicator Q x) = (\<not> (x \<in> Q \<and> \<not> P 1 \<or> x \<notin> Q \<and> \<not> P 0))"
by (auto split: split_indicator)
lemma simple_integral_restrict_space:
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M" "simple_function (restrict_space M \<Omega>) f"
shows "simple_integral (restrict_space M \<Omega>) f = simple_integral M (\<lambda>x. f x * indicator \<Omega> x)"
using simple_function_restrict_space_ereal[THEN iffD1, OF \<Omega>, THEN simple_functionD(1)]
by (auto simp add: space_restrict_space emeasure_restrict_space[OF \<Omega>(1)] le_infI2 simple_integral_def
split: split_indicator split_indicator_asm
intro!: setsum.mono_neutral_cong_left ereal_right_mult_cong[OF refl] arg_cong2[where f=emeasure])
lemma one_not_less_zero[simp]: "\<not> 1 < (0::ereal)"
by (simp add: zero_ereal_def one_ereal_def)
lemma nn_integral_restrict_space:
assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "nn_integral (restrict_space M \<Omega>) f = nn_integral M (\<lambda>x. f x * indicator \<Omega> x)"
proof -
let ?R = "restrict_space M \<Omega>" and ?X = "\<lambda>M f. {s. simple_function M s \<and> s \<le> max 0 \<circ> f \<and> range s \<subseteq> {0 ..< \<infinity>}}"
have "integral\<^sup>S ?R ` ?X ?R f = integral\<^sup>S M ` ?X M (\<lambda>x. f x * indicator \<Omega> x)"
proof (safe intro!: image_eqI)
fix s assume s: "simple_function ?R s" "s \<le> max 0 \<circ> f" "range s \<subseteq> {0..<\<infinity>}"
from s show "integral\<^sup>S (restrict_space M \<Omega>) s = integral\<^sup>S M (\<lambda>x. s x * indicator \<Omega> x)"
by (intro simple_integral_restrict_space) auto
from s show "simple_function M (\<lambda>x. s x * indicator \<Omega> x)"
by (simp add: simple_function_restrict_space_ereal)
from s show "(\<lambda>x. s x * indicator \<Omega> x) \<le> max 0 \<circ> (\<lambda>x. f x * indicator \<Omega> x)"
"\<And>x. s x * indicator \<Omega> x \<in> {0..<\<infinity>}"
by (auto split: split_indicator simp: le_fun_def image_subset_iff)
next
fix s assume s: "simple_function M s" "s \<le> max 0 \<circ> (\<lambda>x. f x * indicator \<Omega> x)" "range s \<subseteq> {0..<\<infinity>}"
then have "simple_function M (\<lambda>x. s x * indicator (\<Omega> \<inter> space M) x)" (is ?s')
by (intro simple_function_mult simple_function_indicator) auto
also have "?s' \<longleftrightarrow> simple_function M (\<lambda>x. s x * indicator \<Omega> x)"
by (rule simple_function_cong) (auto split: split_indicator)
finally show sf: "simple_function (restrict_space M \<Omega>) s"
by (simp add: simple_function_restrict_space_ereal)
from s have s_eq: "s = (\<lambda>x. s x * indicator \<Omega> x)"
by (auto simp add: fun_eq_iff le_fun_def image_subset_iff
split: split_indicator split_indicator_asm
intro: antisym)
show "integral\<^sup>S M s = integral\<^sup>S (restrict_space M \<Omega>) s"
by (subst s_eq) (rule simple_integral_restrict_space[symmetric, OF \<Omega> sf])
show "\<And>x. s x \<in> {0..<\<infinity>}"
using s by (auto simp: image_subset_iff)
from s show "s \<le> max 0 \<circ> f"
by (subst s_eq) (auto simp: image_subset_iff le_fun_def split: split_indicator split_indicator_asm)
qed
then show ?thesis
unfolding nn_integral_def_finite SUP_def by simp
qed
lemma nn_integral_count_space_indicator:
assumes "NO_MATCH (UNIV::'a set) (X::'a set)"
shows "(\<integral>\<^sup>+x. f x \<partial>count_space X) = (\<integral>\<^sup>+x. f x * indicator X x \<partial>count_space UNIV)"
by (simp add: nn_integral_restrict_space[symmetric] restrict_count_space)
lemma nn_integral_count_space_eq:
"(\<And>x. x \<in> A - B \<Longrightarrow> f x = 0) \<Longrightarrow> (\<And>x. x \<in> B - A \<Longrightarrow> f x = 0) \<Longrightarrow>
(\<integral>\<^sup>+x. f x \<partial>count_space A) = (\<integral>\<^sup>+x. f x \<partial>count_space B)"
by (auto simp: nn_integral_count_space_indicator intro!: nn_integral_cong split: split_indicator)
lemma nn_integral_ge_point:
assumes "x \<in> A"
shows "p x \<le> \<integral>\<^sup>+ x. p x \<partial>count_space A"
proof -
from assms have "p x \<le> \<integral>\<^sup>+ x. p x \<partial>count_space {x}"
by(auto simp add: nn_integral_count_space_finite max_def)
also have "\<dots> = \<integral>\<^sup>+ x'. p x' * indicator {x} x' \<partial>count_space A"
using assms by(auto simp add: nn_integral_count_space_indicator indicator_def intro!: nn_integral_cong)
also have "\<dots> \<le> \<integral>\<^sup>+ x. max 0 (p x) \<partial>count_space A"
by(rule nn_integral_mono)(simp add: indicator_def)
also have "\<dots> = \<integral>\<^sup>+ x. p x \<partial>count_space A" by(simp add: nn_integral_def o_def)
finally show ?thesis .
qed
subsubsection \<open>Measure spaces with an associated density\<close>
definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
"density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
lemma
shows sets_density[simp, measurable_cong]: "sets (density M f) = sets M"
and space_density[simp]: "space (density M f) = space M"
by (auto simp: density_def)
(* FIXME: add conversion to simplify space, sets and measurable *)
lemma space_density_imp[measurable_dest]:
"\<And>x M f. x \<in> space (density M f) \<Longrightarrow> x \<in> space M" by auto
lemma
shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
unfolding measurable_def simple_function_def by simp_all
lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
(AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
unfolding density_def by (auto intro!: measure_of_eq nn_integral_cong_AE sets.space_closed)
lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
proof -
have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
by (auto simp: indicator_def)
then show ?thesis
unfolding density_def by (simp add: nn_integral_max_0)
qed
lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
lemma emeasure_density:
assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
shows "emeasure (density M f) A = (\<integral>\<^sup>+ x. f x * indicator A x \<partial>M)"
(is "_ = ?\<mu> A")
unfolding density_def
proof (rule emeasure_measure_of_sigma)
show "sigma_algebra (space M) (sets M)" ..
show "positive (sets M) ?\<mu>"
using f by (auto simp: positive_def intro!: nn_integral_nonneg)
have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
apply (subst nn_integral_max_0[symmetric])
apply (intro ext nn_integral_cong_AE AE_I2)
apply (auto simp: indicator_def)
done
show "countably_additive (sets M) ?\<mu>"
unfolding \<mu>_eq
proof (intro countably_additiveI)
fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
then have "\<And>i. A i \<in> sets M" by auto
then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
by (auto simp: set_eq_iff)
assume disj: "disjoint_family A"
have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^sup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
using f * by (simp add: nn_integral_suminf)
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
by (auto intro!: suminf_cmult_ereal nn_integral_cong_AE)
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
unfolding suminf_indicator[OF disj] ..
finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
qed
qed fact
lemma null_sets_density_iff:
assumes f: "f \<in> borel_measurable M"
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)"
proof -
{ assume "A \<in> sets M"
have eq: "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^sup>+x. max 0 (f x) * indicator A x \<partial>M)"
apply (subst nn_integral_max_0[symmetric])
apply (intro nn_integral_cong)
apply (auto simp: indicator_def)
done
have "(\<integral>\<^sup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
unfolding eq
using f \<open>A \<in> sets M\<close>
by (intro nn_integral_0_iff) auto
also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
using f \<open>A \<in> sets M\<close>
by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
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)"
by (auto simp add: indicator_def max_def split: split_if_asm)
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)" . }
with f show ?thesis
by (simp add: null_sets_def emeasure_density cong: conj_cong)
qed
lemma AE_density:
assumes f: "f \<in> borel_measurable M"
shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
proof
assume "AE x in density M f. P x"
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"
by (auto simp: eventually_ae_filter null_sets_density_iff)
then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
by (rule eventually_elim2) auto
next
fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
by (auto simp: eventually_ae_filter)
then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
"N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
using f by (auto simp: subset_eq intro!: sets.sets_Collect_neg AE_not_in)
show "AE x in density M f. P x"
using ae2
unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
(auto elim: eventually_elim2)
qed
lemma nn_integral_density':
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
shows "integral\<^sup>N (density M f) g = (\<integral>\<^sup>+ x. f x * g x \<partial>M)"
using g proof induct
case (cong u v)
then show ?case
apply (subst nn_integral_cong[OF cong(3)])
apply (simp_all cong: nn_integral_cong)
done
next
case (set A) then show ?case
by (simp add: emeasure_density f)
next
case (mult u c)
moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
ultimately show ?case
using f by (simp add: nn_integral_cmult)
next
case (add u v)
then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
by (simp add: ereal_right_distrib)
with add f show ?case
by (auto simp add: nn_integral_add ereal_zero_le_0_iff intro!: nn_integral_add[symmetric])
next
case (seq U)
from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
by eventually_elim (simp add: SUP_ereal_mult_left seq)
from seq f show ?case
apply (simp add: nn_integral_monotone_convergence_SUP)
apply (subst nn_integral_cong_AE[OF eq])
apply (subst nn_integral_monotone_convergence_SUP_AE)
apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
done
qed
lemma nn_integral_density:
"f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow>
integral\<^sup>N (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
by (subst (1 2) nn_integral_max_0[symmetric])
(auto intro!: nn_integral_cong_AE
simp: measurable_If max_def ereal_zero_le_0_iff nn_integral_density')
lemma density_distr:
assumes [measurable]: "f \<in> borel_measurable N" "X \<in> measurable M N"
shows "density (distr M N X) f = distr (density M (\<lambda>x. f (X x))) N X"
by (intro measure_eqI)
(auto simp add: emeasure_density nn_integral_distr emeasure_distr
split: split_indicator intro!: nn_integral_cong)
lemma emeasure_restricted:
assumes S: "S \<in> sets M" and X: "X \<in> sets M"
shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
proof -
have "emeasure (density M (indicator S)) X = (\<integral>\<^sup>+x. indicator S x * indicator X x \<partial>M)"
using S X by (simp add: emeasure_density)
also have "\<dots> = (\<integral>\<^sup>+x. indicator (S \<inter> X) x \<partial>M)"
by (auto intro!: nn_integral_cong simp: indicator_def)
also have "\<dots> = emeasure M (S \<inter> X)"
using S X by (simp add: sets.Int)
finally show ?thesis .
qed
lemma measure_restricted:
"S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
by (simp add: emeasure_restricted measure_def)
lemma (in finite_measure) finite_measure_restricted:
"S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
by standard (simp add: emeasure_restricted)
lemma emeasure_density_const:
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
by (auto simp: nn_integral_cmult_indicator emeasure_density)
lemma measure_density_const:
"A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real_of_ereal c * measure M A"
by (auto simp: emeasure_density_const measure_def)
lemma density_density_eq:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
density (density M f) g = density M (\<lambda>x. f x * g x)"
by (auto intro!: measure_eqI simp: emeasure_density nn_integral_density ac_simps)
lemma distr_density_distr:
assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
and inv: "\<forall>x\<in>space M. T' (T x) = x"
assumes f: "f \<in> borel_measurable M'"
shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
proof (rule measure_eqI)
fix A assume A: "A \<in> sets ?R"
{ fix x assume "x \<in> space M"
with sets.sets_into_space[OF A]
have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
using T inv by (auto simp: indicator_def measurable_space) }
with A T T' f show "emeasure ?R A = emeasure ?L A"
by (simp add: measurable_comp emeasure_density emeasure_distr
nn_integral_distr measurable_sets cong: nn_integral_cong)
qed simp
lemma density_density_divide:
fixes f g :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
proof -
have "density M g = density M (\<lambda>x. f x * (g x / f x))"
using f g ac by (auto intro!: density_cong measurable_If)
then show ?thesis
using f g by (subst density_density_eq) auto
qed
lemma density_1: "density M (\<lambda>_. 1) = M"
by (intro measure_eqI) (auto simp: emeasure_density)
lemma emeasure_density_add:
assumes X: "X \<in> sets M"
assumes Mf[measurable]: "f \<in> borel_measurable M"
assumes Mg[measurable]: "g \<in> borel_measurable M"
assumes nonnegf: "\<And>x. x \<in> space M \<Longrightarrow> f x \<ge> 0"
assumes nonnegg: "\<And>x. x \<in> space M \<Longrightarrow> g x \<ge> 0"
shows "emeasure (density M f) X + emeasure (density M g) X =
emeasure (density M (\<lambda>x. f x + g x)) X"
using assms
apply (subst (1 2 3) emeasure_density, simp_all) []
apply (subst nn_integral_add[symmetric], simp_all) []
apply (intro nn_integral_cong, simp split: split_indicator)
done
subsubsection \<open>Point measure\<close>
definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
"point_measure A f = density (count_space A) f"
lemma
shows space_point_measure: "space (point_measure A f) = A"
and sets_point_measure: "sets (point_measure A f) = Pow A"
by (auto simp: point_measure_def)
lemma sets_point_measure_count_space[measurable_cong]: "sets (point_measure A f) = sets (count_space A)"
by (simp add: sets_point_measure)
lemma measurable_point_measure_eq1[simp]:
"g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
unfolding point_measure_def by simp
lemma measurable_point_measure_eq2_finite[simp]:
"finite A \<Longrightarrow>
g \<in> measurable M (point_measure A f) \<longleftrightarrow>
(g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
unfolding point_measure_def by (simp add: measurable_count_space_eq2)
lemma simple_function_point_measure[simp]:
"simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
by (simp add: point_measure_def)
lemma emeasure_point_measure:
assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
proof -
have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
using \<open>X \<subseteq> A\<close> by auto
with A show ?thesis
by (simp add: emeasure_density nn_integral_count_space ereal_zero_le_0_iff
point_measure_def indicator_def)
qed
lemma emeasure_point_measure_finite:
"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)"
by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le)
lemma emeasure_point_measure_finite2:
"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)"
by (subst emeasure_point_measure)
(auto dest: finite_subset intro!: setsum.mono_neutral_left simp: less_le)
lemma null_sets_point_measure_iff:
"X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
lemma AE_point_measure:
"(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
unfolding point_measure_def
by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
lemma nn_integral_point_measure:
"finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
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)"
unfolding point_measure_def
apply (subst density_max_0)
apply (subst nn_integral_density)
apply (simp_all add: AE_count_space nn_integral_density)
apply (subst nn_integral_count_space )
apply (auto intro!: setsum.cong simp: max_def ereal_zero_less_0_iff)
apply (rule finite_subset)
prefer 2
apply assumption
apply auto
done
lemma nn_integral_point_measure_finite:
"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>
integral\<^sup>N (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
by (subst nn_integral_point_measure) (auto intro!: setsum.mono_neutral_left simp: less_le)
subsubsection \<open>Uniform measure\<close>
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
lemma
shows sets_uniform_measure[simp, measurable_cong]: "sets (uniform_measure M A) = sets M"
and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
by (auto simp: uniform_measure_def)
lemma emeasure_uniform_measure[simp]:
assumes A: "A \<in> sets M" and B: "B \<in> sets M"
shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
proof -
from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^sup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
intro!: nn_integral_cong)
also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
using A B
by (subst nn_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
finally show ?thesis .
qed
lemma measure_uniform_measure[simp]:
assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
lemma AE_uniform_measureI:
"A \<in> sets M \<Longrightarrow> (AE x in M. x \<in> A \<longrightarrow> P x) \<Longrightarrow> (AE x in uniform_measure M A. P x)"
unfolding uniform_measure_def by (auto simp: AE_density)
lemma emeasure_uniform_measure_1:
"emeasure M S \<noteq> 0 \<Longrightarrow> emeasure M S \<noteq> \<infinity> \<Longrightarrow> emeasure (uniform_measure M S) S = 1"
by (subst emeasure_uniform_measure)
(simp_all add: emeasure_nonneg emeasure_neq_0_sets)
lemma nn_integral_uniform_measure:
assumes f[measurable]: "f \<in> borel_measurable M" and "\<And>x. 0 \<le> f x" and S[measurable]: "S \<in> sets M"
shows "(\<integral>\<^sup>+x. f x \<partial>uniform_measure M S) = (\<integral>\<^sup>+x. f x * indicator S x \<partial>M) / emeasure M S"
proof -
{ assume "emeasure M S = \<infinity>"
then have ?thesis
by (simp add: uniform_measure_def nn_integral_density f) }
moreover
{ assume [simp]: "emeasure M S = 0"
then have ae: "AE x in M. x \<notin> S"
using sets.sets_into_space[OF S]
by (subst AE_iff_measurable[OF _ refl]) (simp_all add: subset_eq cong: rev_conj_cong)
from ae have "(\<integral>\<^sup>+ x. indicator S x * f x / 0 \<partial>M) = 0"
by (subst nn_integral_0_iff_AE) auto
moreover from ae have "(\<integral>\<^sup>+ x. f x * indicator S x \<partial>M) = 0"
by (subst nn_integral_0_iff_AE) auto
ultimately have ?thesis
by (simp add: uniform_measure_def nn_integral_density f) }
moreover
{ assume "emeasure M S \<noteq> 0" "emeasure M S \<noteq> \<infinity>"
moreover then have "0 < emeasure M S"
by (simp add: emeasure_nonneg less_le)
ultimately have ?thesis
unfolding uniform_measure_def
apply (subst nn_integral_density)
apply (auto simp: f nn_integral_divide intro!: zero_le_divide_ereal)
apply (simp add: mult.commute)
done }
ultimately show ?thesis by blast
qed
lemma AE_uniform_measure:
assumes "emeasure M A \<noteq> 0" "emeasure M A < \<infinity>"
shows "(AE x in uniform_measure M A. P x) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> P x)"
proof -
have "A \<in> sets M"
using \<open>emeasure M A \<noteq> 0\<close> by (metis emeasure_notin_sets)
moreover have "\<And>x. 0 < indicator A x / emeasure M A \<longleftrightarrow> x \<in> A"
using emeasure_nonneg[of M A] assms
by (cases "emeasure M A") (auto split: split_indicator simp: one_ereal_def)
ultimately show ?thesis
unfolding uniform_measure_def by (simp add: AE_density)
qed
subsubsection \<open>Null measure\<close>
lemma null_measure_eq_density: "null_measure M = density M (\<lambda>_. 0)"
by (intro measure_eqI) (simp_all add: emeasure_density)
lemma nn_integral_null_measure[simp]: "(\<integral>\<^sup>+x. f x \<partial>null_measure M) = 0"
by (auto simp add: nn_integral_def simple_integral_def SUP_constant bot_ereal_def le_fun_def
intro!: exI[of _ "\<lambda>x. 0"])
lemma density_null_measure[simp]: "density (null_measure M) f = null_measure M"
proof (intro measure_eqI)
fix A show "emeasure (density (null_measure M) f) A = emeasure (null_measure M) A"
by (simp add: density_def) (simp only: null_measure_def[symmetric] emeasure_null_measure)
qed simp
subsubsection \<open>Uniform count measure\<close>
definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
lemma
shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
lemma sets_uniform_count_measure_count_space[measurable_cong]:
"sets (uniform_count_measure A) = sets (count_space A)"
by (simp add: sets_uniform_count_measure)
lemma emeasure_uniform_count_measure:
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
by (simp add: emeasure_point_measure_finite uniform_count_measure_def)
lemma measure_uniform_count_measure:
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
by (simp add: emeasure_point_measure_finite uniform_count_measure_def measure_def)
lemma space_uniform_count_measure_empty_iff [simp]:
"space (uniform_count_measure X) = {} \<longleftrightarrow> X = {}"
by(simp add: space_uniform_count_measure)
lemma sets_uniform_count_measure_eq_UNIV [simp]:
"sets (uniform_count_measure UNIV) = UNIV \<longleftrightarrow> True"
"UNIV = sets (uniform_count_measure UNIV) \<longleftrightarrow> True"
by(simp_all add: sets_uniform_count_measure)
subsubsection \<open>Scaled measure\<close>
lemma nn_integral_scale_measure':
assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x"
shows "nn_integral (scale_measure r M) f = max 0 r * nn_integral M f"
using f
proof induction
case (cong f g)
thus ?case
by(simp add: cong.hyps space_scale_measure cong: nn_integral_cong_simp)
next
case (mult f c)
thus ?case
by(simp add: nn_integral_cmult max_def mult.assoc mult.left_commute)
next
case (add f g)
thus ?case
by(simp add: nn_integral_add ereal_right_distrib nn_integral_nonneg)
next
case (seq U)
thus ?case
by(simp add: nn_integral_monotone_convergence_SUP SUP_ereal_mult_left nn_integral_nonneg)
qed simp
lemma nn_integral_scale_measure:
"f \<in> borel_measurable M \<Longrightarrow> nn_integral (scale_measure r M) f = max 0 r * nn_integral M f"
by(subst (1 2) nn_integral_max_0[symmetric])(rule nn_integral_scale_measure', simp_all)
end