tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/Probability/Lebesgue_Integration.thy
Author: Johannes Hölzl, TU München
Author: Armin Heller, TU München
*)
header {*Lebesgue Integration*}
theory Lebesgue_Integration
imports Measure_Space Borel_Space
begin
lemma tendsto_real_max:
fixes x y :: real
assumes "(X ---> x) net"
assumes "(Y ---> y) net"
shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
proof -
have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
by (auto split: split_max simp: field_simps)
show ?thesis
unfolding *
by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
qed
lemma measurable_sets2[intro]:
assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
and "A \<in> sets M'" "B \<in> sets M''"
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 by (auto intro: measurable_sets)
qed
section "Simple function"
text {*
Our simple functions are not restricted to positive real numbers. Instead
they are just functions with a finite range and are measurable when singleton
sets are measurable.
*}
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 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_cong2)
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 borel_measurable_simple_function[measurable_dest]:
assumes "simple_function M f"
shows "f \<in> borel_measurable M"
proof (rule borel_measurableI)
fix S
let ?I = "f ` (f -` S \<inter> space M)"
have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
have "finite ?I"
using assms unfolding simple_function_def
using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
hence "?U \<in> sets M"
apply (rule sets.finite_UN)
using assms unfolding simple_function_def by auto
thus "f -` S \<inter> space M \<in> sets M" unfolding * .
qed
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_borel_measurable:
fixes f :: "'a \<Rightarrow> ereal"
shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
using simple_function_borel_measurable[of f] borel_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_compose)
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 `x \<in> space M` 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
fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))"
by (auto intro!: simple_function_compose1[OF sf])
lemma
fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
by (auto intro!: 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 natfloor (real (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 "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
unfolding real_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 (natfloor (real (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 intro: simple_function_ereal simple_function_div)
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> natfloor (real (u x) * 2 ^ Suc i)"
by (cases "u x") (auto intro!: le_natfloor)
next
assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
then show "natfloor (real (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 "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
by simp
also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
proof cases
assume "0 \<le> u x" then show ?thesis
by (intro le_mult_natfloor)
next
assume "\<not> 0 \<le> u x" then show ?thesis
by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
qed
also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
by (simp add: ac_simps)
finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (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 real_natfloor_le natfloor_neg
mult_nonpos_nonneg mult_nonneg_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 simp: divide_nonneg_pos)
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: inverse_eq_divide field_simps)
then have "r * 2^max N m < p * 2^max N m - 1" by simp
moreover
have "real (natfloor (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 mult_nonneg_nonneg split: split_if_asm)
then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
by (metis real_natfloor_gt_diff_one less_le_trans)
ultimately show False by auto
qed
then show "max 0 (u x) \<le> y" using real ux by simp
qed (insert `0 \<le> y`, auto)
qed
qed (auto simp: divide_nonneg_pos)
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> 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])
qed (auto intro!: add mult set simple_functionD u setsum_nonneg
simple_function_setsum)
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> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<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> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<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. P (U i)) \<Longrightarrow> incseq U \<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)
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 `simple_function M (U i)` nn
by (induct rule: simple_function_induct_nn)
(auto intro: set mult add cong 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
section "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"
shows "integral\<^sup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * (emeasure M) A)"
(is "_ = setsum _ (?p ` space M)")
proof-
let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)"
let ?SIGMA = "Sigma (f`space M) ?sub"
have [intro]:
"finite (f ` space M)"
"finite (g ` space M)"
using assms unfolding simple_function_def by simp_all
{ fix A
have "?p ` (A \<inter> space M) \<subseteq>
(\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
by auto
hence "finite (?p ` (A \<inter> space M))"
by (rule finite_subset) auto }
note this[intro, simp]
note sets = simple_function_measurable2[OF f g]
{ fix x assume "x \<in> space M"
have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))"
by (subst setsum_emeasure) (auto simp: disjoint_family_on_def) }
hence "integral\<^sup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)"
unfolding simple_integral_def using f sets
by (subst setsum_Sigma[symmetric])
(auto intro!: setsum_cong setsum_ereal_right_distrib)
also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)"
proof -
have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
= (\<lambda>x. (f x, ?p x)) ` space M"
proof safe
fix x assume "x \<in> space M"
thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
by (auto intro!: image_eqI[of _ _ "?p x"])
qed auto
thus ?thesis
apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
apply (rule_tac x="xa" in image_eqI)
by simp_all
qed
finally show ?thesis .
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 -
{ fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
assume "x \<in> space M"
hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
"(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
by auto }
with assms show ?thesis
unfolding
simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
simple_function_partition[OF f g]
simple_function_partition[OF g f]
by (subst (3) Int_commute)
(auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
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 -
note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)]
{ fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
assume "x \<in> space M"
hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
by auto }
with assms show ?thesis
unfolding simple_function_partition[OF mult f(1)]
simple_function_partition[OF f(1) mult]
by (subst setsum_ereal_right_distrib)
(auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
qed
lemma simple_integral_mono_AE:
assumes f: "simple_function M f" and g: "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 ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
"\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
show ?thesis
unfolding *
simple_function_partition[OF f g]
simple_function_partition[OF g f]
proof (safe intro!: setsum_mono)
fix x assume "x \<in> space M"
then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)"
proof (cases "f x \<le> g x")
case True then show ?thesis
using * assms(1,2)[THEN simple_functionD(2)]
by (auto intro!: ereal_mult_right_mono)
next
case False
obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0"
using mono by (auto elim!: AE_E)
have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
moreover have "?S x \<in> sets M" using assms
by (rule_tac sets.Int) (auto intro!: simple_functionD)
ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N"
using `N \<in> sets M` by (auto intro!: emeasure_mono)
moreover have "0 \<le> (emeasure M) (?S x)"
using assms(1,2)[THEN simple_functionD(2)] by auto
ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto
then show ?thesis by simp
qed
qed
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 \<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 (cases "A = space M")
case True
then have "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) = integral\<^sup>S M f"
by (auto intro!: simple_integral_cong)
with True show ?thesis by (simp add: simple_integral_def)
next
assume "A \<noteq> space M"
then obtain x where x: "x \<in> space M" "x \<notin> A" using sets.sets_into_space[OF assms(1)] by auto
have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
proof safe
fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
next
fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
using sets.sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
next
show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
qed
have *: "(\<integral>\<^sup>Sx. f x * indicator A x \<partial>M) =
(\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
unfolding simple_integral_def I
proof (rule setsum_mono_zero_cong_left)
show "finite (f ` space M \<union> {0})"
using assms(2) unfolding simple_function_def by auto
show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
using sets.sets_into_space[OF assms(1)] by auto
have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
by (auto simp: image_iff)
thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto
next
fix x assume "x \<in> f`A \<union> {0}"
hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
by (auto simp: indicator_def split: split_if_asm)
thus "x * (emeasure M) (?I -` {x} \<inter> space M) =
x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
qed
show ?thesis unfolding *
using assms(2) unfolding simple_function_def
by (auto intro!: setsum_mono_zero_cong_right)
qed
lemma simple_integral_indicator_only[simp]:
assumes "A \<in> sets M"
shows "integral\<^sup>S M (indicator A) = emeasure M A"
proof cases
assume "space M = {}" hence "A = {}" using sets.sets_into_space[OF assms] by auto
thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
next
assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
thus ?thesis
using simple_integral_indicator[OF assms simple_function_const[of _ 1]]
using sets.sets_into_space[OF assms]
by (auto intro!: arg_cong[where f="(emeasure M)"])
qed
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 `N \<in> null_sets M` 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_positive:
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
section "Continuous positive integration"
definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^sup>P") where
"integral\<^sup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^sup>S M g)"
syntax
"_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^sup>+ _. _ \<partial>_" [60,61] 110)
translations
"\<integral>\<^sup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
lemma positive_integral_positive:
"0 \<le> integral\<^sup>P M f"
by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
lemma positive_integral_not_MInfty[simp]: "integral\<^sup>P M f \<noteq> -\<infinity>"
using positive_integral_positive[of M f] by auto
lemma positive_integral_def_finite:
"integral\<^sup>P 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 "_ = SUPR ?A ?f")
unfolding positive_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> SUPR ?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 "SUPR ?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 `0 \<le> ?y` `?g ?y \<in> ?A` gM
by (subst simple_integral_cmult_indicator) auto
also have "\<dots> \<le> integral\<^sup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
by (intro simple_integral_mono) auto
finally show "\<exists>i\<in>?A. real n \<le> integral\<^sup>S M i"
using `?g ?y \<in> ?A` by blast
qed
then show ?thesis by simp
qed
qed (auto intro: SUP_upper)
lemma positive_integral_mono_AE:
assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^sup>P M u \<le> integral\<^sup>P M v"
unfolding positive_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 positive_integral_mono:
"(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^sup>P M u \<le> integral\<^sup>P M v"
by (auto intro: positive_integral_mono_AE)
lemma positive_integral_cong_AE:
"AE x in M. u x = v x \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
by (auto simp: eq_iff intro!: positive_integral_mono_AE)
lemma positive_integral_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^sup>P M u = integral\<^sup>P M v"
by (auto intro: positive_integral_cong_AE)
lemma positive_integral_eq_simple_integral:
assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^sup>P 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>P M ?f \<le> integral\<^sup>S M ?f" using f'
by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
moreover have "integral\<^sup>S M ?f \<le> integral\<^sup>P M ?f"
unfolding positive_integral_def
using f' by (auto intro!: SUP_upper)
ultimately show ?thesis
by (simp cong: positive_integral_cong simple_integral_cong)
qed
lemma positive_integral_eq_simple_integral_AE:
assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^sup>P 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>P M f = integral\<^sup>S M (\<lambda>x. max 0 (f x))"
by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
add: positive_integral_eq_simple_integral)
with assms show ?thesis
by (auto intro!: simple_integral_cong_AE split: split_max)
qed
lemma positive_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>P M (f i))" (is "_ \<le> ?S")
proof (rule ereal_le_mult_one_interval)
have "0 \<le> (SUP i. integral\<^sup>P M (f i))"
using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
then show "(SUP i. integral\<^sup>P 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 `simple_function M u` 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 `incseq f`[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 `x \<in> space M` f(3) by simp
next
assume "u x \<noteq> 0"
with `a < 1` u_range[OF `x \<in> space M`]
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 `x \<in> space M` 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 `simple_function M u`]
proof (subst SUPR_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 SUPR_ereal_cmult[symmetric])
qed
moreover
have "a * (SUP i. integral\<^sup>S M (?uB i)) \<le> ?S"
apply (subst SUPR_ereal_cmult[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 `simple_function M u` u_range
by (subst simple_integral_mult) (auto split: split_indicator)
also have "\<dots> \<le> integral\<^sup>P M (f i)"
proof -
have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
show ?thesis using f(3) * u_range `0 < a`
by (subst positive_integral_eq_simple_integral[symmetric])
(auto intro!: positive_integral_mono split: split_indicator)
qed
finally show "a * integral\<^sup>S M (?uB i) \<le> integral\<^sup>P M (f i)"
by auto
next
fix i show "0 \<le> \<integral>\<^sup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
by (intro simple_integral_positive) (auto split: split_indicator)
qed (insert `0 < a`, auto)
ultimately show "a * integral\<^sup>S M u \<le> ?S" by simp
qed
lemma incseq_positive_integral:
assumes "incseq f" shows "incseq (\<lambda>i. integral\<^sup>P 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 positive_integral_mono)
qed
text {* Beppo-Levi monotone convergence theorem *}
lemma positive_integral_monotone_convergence_SUP:
assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
shows "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^sup>P M (f i))"
proof (rule antisym)
show "(SUP j. integral\<^sup>P M (f j)) \<le> (\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M)"
by (auto intro!: SUP_least SUP_upper positive_integral_mono)
next
show "(\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^sup>P M (f j))"
unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. 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. f i x)" "range g \<subseteq> {0..<\<infinity>}"
then have "\<And>x. 0 \<le> (SUP i. 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>P M (f j))"
by (intro positive_integral_SUP_approx[OF f g _ g'])
(auto simp: le_fun_def max_def)
qed
qed
lemma positive_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>P 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!: positive_integral_cong_AE)
also have "\<dots> = (SUP i. (\<integral>\<^sup>+ x. ?f i x \<partial>M))"
proof (rule positive_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
fix x show "0 \<le> ?f i x"
using N(1) by 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="SUPR UNIV"] positive_integral_cong_AE ext)
finally show ?thesis .
qed
lemma positive_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>P M (f i))"
using f[unfolded incseq_Suc_iff le_fun_def]
by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
auto
lemma positive_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 positive_integral_monotone_convergence_SUP[OF f(1)
f(3)[THEN borel_measurable_simple_function] f(2)]
by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
lemma positive_integral_max_0:
"(\<integral>\<^sup>+x. max 0 (f x) \<partial>M) = integral\<^sup>P M f"
by (simp add: le_fun_def positive_integral_def)
lemma positive_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>P M f = integral\<^sup>P M g"
proof -
have "integral\<^sup>P M (\<lambda>x. max 0 (f x)) = integral\<^sup>P M (\<lambda>x. max 0 (g x))"
proof (intro positive_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: positive_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 "SUPR _ ?F = SUPR _ ?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 positive_integral_monotone_convergence_simple)
also have "\<dots> = (\<integral>\<^sup>+x. (SUP i. g i x) \<partial>M)"
unfolding eq[THEN positive_integral_cong_AE] ..
also have "\<dots> = (SUP i. ?G i)"
using g by (rule positive_integral_monotone_convergence_simple[symmetric])
finally show ?thesis by simp
qed
lemma positive_integral_const[simp]:
"0 \<le> c \<Longrightarrow> (\<integral>\<^sup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
by (subst positive_integral_eq_simple_integral) auto
lemma positive_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>P M f + integral\<^sup>P M g"
(is "integral\<^sup>P M ?L = _")
proof -
from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
note v = positive_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 = positive_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 `0 \<le> a`
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 `0 \<le> a` by (auto simp: simple_integral_positive)
{ 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 `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg)
{ fix x
{ fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
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 `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`])
(auto intro!: SUPR_ereal_add
simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) }
then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg)
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) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR 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 SUPR_ereal_cmult[symmetric, OF pos(1) `0 \<le> a`])
apply (subst SUPR_ereal_add[symmetric, OF inc not_MInf]) .
then show ?thesis by (simp add: positive_integral_max_0)
qed
lemma positive_integral_cmult:
assumes f: "f \<in> borel_measurable M" "0 \<le> c"
shows "(\<integral>\<^sup>+ x. c * f x \<partial>M) = c * integral\<^sup>P M f"
proof -
have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
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: positive_integral_max_0)
then show ?thesis
using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
by (auto simp: positive_integral_max_0)
qed
lemma positive_integral_multc:
assumes "f \<in> borel_measurable M" "0 \<le> c"
shows "(\<integral>\<^sup>+ x. f x * c \<partial>M) = integral\<^sup>P M f * c"
unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
lemma positive_integral_indicator[simp]:
"A \<in> sets M \<Longrightarrow> (\<integral>\<^sup>+ x. indicator A x\<partial>M) = (emeasure M) A"
by (subst positive_integral_eq_simple_integral)
(auto simp: simple_integral_indicator)
lemma positive_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 positive_integral_eq_simple_integral)
(auto simp: simple_function_indicator simple_integral_indicator)
lemma positive_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 positive_integral_cong) (simp split: split_indicator)
also have "\<dots> = emeasure M (A \<inter> space M)"
by simp
finally show ?thesis .
qed
lemma positive_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>P M f + integral\<^sup>P 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 simp: ereal_add_nonneg_nonneg)
have "(\<integral>\<^sup>+ x. f x + g x \<partial>M) = (\<integral>\<^sup>+ x. max 0 (f x + g x) \<partial>M)"
by (simp add: positive_integral_max_0)
also have "\<dots> = (\<integral>\<^sup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
unfolding ae[THEN positive_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 positive_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: positive_integral_max_0)
qed
lemma positive_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>P 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 positive_integral_add[OF this]
show ?case using insert by auto
qed simp
qed simp
lemma positive_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 `A \<in> sets M` u by auto
hence "(emeasure M) ?A = (\<integral>\<^sup>+ x. indicator ?A x \<partial>M)"
using positive_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!: positive_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!: positive_integral_cmult simp: ereal_zero_le_0_iff)
finally show ?thesis .
qed
lemma positive_integral_noteq_infinite:
assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
and "integral\<^sup>P 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 positive_integral_cmult_indicator) auto
also have "\<dots> \<le> integral\<^sup>P M g"
using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
finally show False using `integral\<^sup>P M g \<noteq> \<infinity>` by auto
qed
lemma positive_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>P 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>P M f - integral\<^sup>P 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 positive_integral_noteq_infinite[OF g fin] assms by auto
then have **: "integral\<^sup>P M f = (\<integral>\<^sup>+x. f x - g x \<partial>M) + integral\<^sup>P M g"
unfolding positive_integral_add[OF diff g, symmetric]
by (rule positive_integral_cong_AE)
show ?thesis unfolding **
using fin positive_integral_positive[of M g]
by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. f x - g x \<partial>M" "integral\<^sup>P M g"]) auto
qed
lemma positive_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>P 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>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^sup>P M (f i))"
using positive_integral_positive by (rule suminf_ereal_eq_SUPR)
also have "\<dots> = (SUP n. \<integral>\<^sup>+x. (\<Sum>i<n. f i x) \<partial>M)"
unfolding positive_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 positive_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 positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUPR)
finally show ?thesis by simp
qed
text {* Fatou's lemma: convergence theorem on limes inferior *}
lemma positive_integral_lim_INF:
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>P 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_SUPR_INFI using pos u
by (intro positive_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>P M (u n))"
unfolding liminf_SUPR_INFI
by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
finally show ?thesis .
qed
lemma positive_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 positive_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 positive_integral_0_iff:
assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
shows "integral\<^sup>P 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>P M u"
by (auto intro!: positive_integral_cong simp: indicator_def)
show ?thesis
proof
assume "(emeasure M) ?A = 0"
with positive_integral_null_set[of ?A M u] u
show "integral\<^sup>P 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>P 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 positive_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 `0 \<le> u x` 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 `0 < u x` 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 `0 < r` by auto
hence "1 \<le> real j * r" using real `0 < r` by auto
thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
qed (insert `0 < u x`, 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 positive_integral_0_iff_AE:
assumes u: "u \<in> borel_measurable M"
shows "integral\<^sup>P 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 positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
have "integral\<^sup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
unfolding positive_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_positive_integral:
"{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^sup>P M (indicator {x. \<not> P x}) = 0"
by (subst positive_integral_0_iff_AE) (auto simp: one_ereal_def zero_ereal_def
sets.sets_Collect_neg indicator_def[abs_def] measurable_If)
lemma positive_integral_const_If:
"(\<integral>\<^sup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
lemma positive_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>P N f = integral\<^sup>P 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: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
done
qed
lemma positive_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 `x \<in> space M` 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: positive_integral_cong)
also have "\<dots> = (\<Sum>i. emeasure M (F i))"
by (simp add: positive_integral_suminf)
finally show ?thesis
by (simp add: F_def)
qed
section "Lebesgue Integral"
definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
"integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
lemma borel_measurable_integrable[measurable_dest]:
"integrable M f \<Longrightarrow> f \<in> borel_measurable M"
by (auto simp: integrable_def)
lemma integrableD[dest]:
assumes "integrable M f"
shows "f \<in> borel_measurable M" "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
using assms unfolding integrable_def by auto
definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^sup>L") where
"integral\<^sup>L M f = real ((\<integral>\<^sup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^sup>+ x. ereal (- f x) \<partial>M))"
syntax
"_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
translations
"\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
lemma integrableE:
assumes "integrable M f"
obtains r q where
"(\<integral>\<^sup>+x. ereal (f x)\<partial>M) = ereal r"
"(\<integral>\<^sup>+x. ereal (-f x)\<partial>M) = ereal q"
"f \<in> borel_measurable M" "integral\<^sup>L M f = r - q"
using assms unfolding integrable_def lebesgue_integral_def
using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
by (cases rule: ereal2_cases[of "(\<integral>\<^sup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^sup>+x. ereal (f x)\<partial>M)"]) auto
lemma integral_cong:
assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "integral\<^sup>L M f = integral\<^sup>L M g"
using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
lemma integral_cong_AE:
assumes cong: "AE x in M. f x = g x"
shows "integral\<^sup>L M f = integral\<^sup>L M g"
proof -
have *: "AE x in M. ereal (f x) = ereal (g x)"
"AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
show ?thesis
unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
qed
lemma integrable_cong_AE:
assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
assumes "AE x in M. f x = g x"
shows "integrable M f = integrable M g"
proof -
have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (g x) \<partial>M)"
"(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^sup>+ x. ereal (- g x) \<partial>M)"
using assms by (auto intro!: positive_integral_cong_AE)
with assms show ?thesis
by (auto simp: integrable_def)
qed
lemma integrable_cong:
"(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
lemma integral_mono_AE:
assumes fg: "integrable M f" "integrable M g"
and mono: "AE t in M. f t \<le> g t"
shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
proof -
have "AE x in M. ereal (f x) \<le> ereal (g x)"
using mono by auto
moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
using mono by auto
ultimately show ?thesis using fg
by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
simp: positive_integral_positive lebesgue_integral_def diff_minus)
qed
lemma integral_mono:
assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
using assms by (auto intro: integral_mono_AE)
lemma positive_integral_eq_integral:
assumes f: "integrable M f"
assumes nonneg: "AE x in M. 0 \<le> f x"
shows "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) = integral\<^sup>L M f"
proof -
have "(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+ x. 0 \<partial>M)"
using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
with f positive_integral_positive show ?thesis
by (cases "\<integral>\<^sup>+ x. ereal (f x) \<partial>M")
(auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
qed
lemma integral_eq_positive_integral:
assumes f: "\<And>x. 0 \<le> f x"
shows "integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
proof -
{ fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
then have "0 = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
also have "\<dots> = (\<integral>\<^sup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
finally show ?thesis
unfolding lebesgue_integral_def by simp
qed
lemma integral_minus[intro, simp]:
assumes "integrable M f"
shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
using assms by (auto simp: integrable_def lebesgue_integral_def)
lemma integral_minus_iff[simp]:
"integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
proof
assume "integrable M (\<lambda>x. - f x)"
then have "integrable M (\<lambda>x. - (- f x))"
by (rule integral_minus)
then show "integrable M f" by simp
qed (rule integral_minus)
lemma integral_of_positive_diff:
assumes integrable: "integrable M u" "integrable M v"
and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
shows "integrable M f" and "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
proof -
let ?f = "\<lambda>x. max 0 (ereal (f x))"
let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
let ?u = "\<lambda>x. max 0 (ereal (u x))"
let ?v = "\<lambda>x. max 0 (ereal (v x))"
from borel_measurable_diff[of u M v] integrable
have f_borel: "?f \<in> borel_measurable M" and
mf_borel: "?mf \<in> borel_measurable M" and
v_borel: "?v \<in> borel_measurable M" and
u_borel: "?u \<in> borel_measurable M" and
"f \<in> borel_measurable M"
by (auto simp: f_def[symmetric] integrable_def)
have "(\<integral>\<^sup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^sup>P M ?u"
using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
moreover have "(\<integral>\<^sup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^sup>P M ?v"
using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
ultimately show f: "integrable M f"
using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
by (auto simp: integrable_def f_def positive_integral_max_0)
have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
unfolding f_def using pos by (simp split: split_max)
then have "(\<integral>\<^sup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^sup>+ x. ?v x + ?f x \<partial>M)" by simp
then have "real (integral\<^sup>P M ?u + integral\<^sup>P M ?mf) =
real (integral\<^sup>P M ?v + integral\<^sup>P M ?f)"
using positive_integral_add[OF u_borel _ mf_borel]
using positive_integral_add[OF v_borel _ f_borel]
by auto
then show "integral\<^sup>L M f = integral\<^sup>L M u - integral\<^sup>L M v"
unfolding positive_integral_max_0
unfolding pos[THEN integral_eq_positive_integral]
using integrable f by (auto elim!: integrableE)
qed
lemma integral_linear:
assumes "integrable M f" "integrable M g" and "0 \<le> a"
shows "integrable M (\<lambda>t. a * f t + g t)"
and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^sup>L M f + integral\<^sup>L M g" (is ?EQ)
proof -
let ?f = "\<lambda>x. max 0 (ereal (f x))"
let ?g = "\<lambda>x. max 0 (ereal (g x))"
let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
let ?mg = "\<lambda>x. max 0 (ereal (- g x))"
let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)"
let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)"
from assms have linear:
"(\<integral>\<^sup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^sup>P M ?f + integral\<^sup>P M ?g"
"(\<integral>\<^sup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^sup>P M ?mf + integral\<^sup>P M ?mg"
by (auto intro!: positive_integral_linear simp: integrable_def)
have *: "(\<integral>\<^sup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^sup>+x. ereal (- ?n x) \<partial>M) = 0"
using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
"\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
have "integrable M ?p" "integrable M ?n"
"\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
using linear assms unfolding integrable_def ** *
by (auto simp: positive_integral_max_0)
note diff = integral_of_positive_diff[OF this]
show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
from assms linear show ?EQ
unfolding diff(2) ** positive_integral_max_0
unfolding lebesgue_integral_def *
by (auto elim!: integrableE simp: field_simps)
qed
lemma integral_add[simp, intro]:
assumes "integrable M f" "integrable M g"
shows "integrable M (\<lambda>t. f t + g t)"
and "(\<integral> t. f t + g t \<partial>M) = integral\<^sup>L M f + integral\<^sup>L M g"
using assms integral_linear[where a=1] by auto
lemma integral_zero[simp, intro]:
shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
unfolding integrable_def lebesgue_integral_def
by auto
lemma lebesgue_integral_uminus:
"(\<integral>x. - f x \<partial>M) = - integral\<^sup>L M f"
unfolding lebesgue_integral_def by simp
lemma lebesgue_integral_cmult_nonneg:
assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
proof -
{ have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (f x)))) =
real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
using f `0 \<le> c` by (subst positive_integral_cmult) auto
also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
finally have "real (integral\<^sup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (f x)))"
by (simp add: positive_integral_max_0) }
moreover
{ have "real (ereal c * integral\<^sup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
real (integral\<^sup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
using f `0 \<le> c` by (subst positive_integral_cmult) auto
also have "\<dots> = real (integral\<^sup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
finally have "real (integral\<^sup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^sup>P M (\<lambda>x. ereal (- f x)))"
by (simp add: positive_integral_max_0) }
ultimately show ?thesis
by (simp add: lebesgue_integral_def field_simps)
qed
lemma lebesgue_integral_cmult:
assumes f: "f \<in> borel_measurable M"
shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^sup>L M f"
proof (cases rule: linorder_le_cases)
assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
next
assume "c \<le> 0"
with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
show ?thesis
by (simp add: lebesgue_integral_def)
qed
lemma lebesgue_integral_multc:
"f \<in> borel_measurable M \<Longrightarrow> (\<integral>x. f x * c \<partial>M) = integral\<^sup>L M f * c"
using lebesgue_integral_cmult[of f M c] by (simp add: ac_simps)
lemma integral_multc:
"integrable M f \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
by (simp add: lebesgue_integral_multc)
lemma integral_cmult[simp, intro]:
assumes "integrable M f"
shows "integrable M (\<lambda>t. a * f t)" (is ?P)
and "(\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f" (is ?I)
proof -
have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^sup>L M f"
proof (cases rule: le_cases)
assume "0 \<le> a" show ?thesis
using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
by simp
next
assume "a \<le> 0" hence "0 \<le> - a" by auto
have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
integral_minus(1)[of M "\<lambda>t. - a * f t"]
unfolding * integral_zero by simp
qed
thus ?P ?I by auto
qed
lemma integral_diff[simp, intro]:
assumes f: "integrable M f" and g: "integrable M g"
shows "integrable M (\<lambda>t. f t - g t)"
and "(\<integral> t. f t - g t \<partial>M) = integral\<^sup>L M f - integral\<^sup>L M g"
using integral_add[OF f integral_minus(1)[OF g]]
unfolding diff_minus integral_minus(2)[OF g]
by auto
lemma integral_indicator[simp, intro]:
assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
shows "integral\<^sup>L M (indicator A) = real (emeasure M A)" (is ?int)
and "integrable M (indicator A)" (is ?able)
proof -
from `A \<in> sets M` have *:
"\<And>x. ereal (indicator A x) = indicator A x"
"(\<integral>\<^sup>+x. ereal (- indicator A x) \<partial>M) = 0"
by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
show ?int ?able
using assms unfolding lebesgue_integral_def integrable_def
by (auto simp: *)
qed
lemma integral_cmul_indicator:
assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
proof -
show ?P
proof (cases "c = 0")
case False with assms show ?thesis by simp
qed simp
show ?I
proof (cases "c = 0")
case False with assms show ?thesis by simp
qed simp
qed
lemma integral_setsum[simp, intro]:
assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^sup>L M (f i))" (is "?int S")
and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
proof -
have "?int S \<and> ?I S"
proof (cases "finite S")
assume "finite S"
from this assms show ?thesis by (induct S) simp_all
qed simp
thus "?int S" and "?I S" by auto
qed
lemma integrable_bound:
assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x"
assumes borel: "g \<in> borel_measurable M"
shows "integrable M g"
proof -
have "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
by (auto intro!: positive_integral_mono)
also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono_AE)
also have "\<dots> < \<infinity>"
using `integrable M f` unfolding integrable_def by auto
finally have pos: "(\<integral>\<^sup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
have "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
by (auto intro!: positive_integral_mono)
also have "\<dots> \<le> (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
using f by (auto intro!: positive_integral_mono_AE)
also have "\<dots> < \<infinity>"
using `integrable M f` unfolding integrable_def by auto
finally have neg: "(\<integral>\<^sup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
from neg pos borel show ?thesis
unfolding integrable_def by auto
qed
lemma integrable_abs:
assumes f[measurable]: "integrable M f"
shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
proof -
from assms have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
"\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
with assms show ?thesis
by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
qed
lemma integral_subalgebra:
assumes borel: "f \<in> borel_measurable N"
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 "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
and "integral\<^sup>L N f = integral\<^sup>L M f" (is ?I)
proof -
have "(\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M)"
"(\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
using borel by (auto intro!: positive_integral_subalgebra N)
moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
using assms unfolding measurable_def by auto
ultimately show ?P ?I
by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
qed
lemma lebesgue_integral_nonneg:
assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^sup>L M f"
proof -
have "(\<integral>\<^sup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^sup>+x. 0 \<partial>M)"
using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
then show ?thesis
by (auto simp: lebesgue_integral_def positive_integral_max_0
intro!: real_of_ereal_pos positive_integral_positive)
qed
lemma integrable_abs_iff:
"f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
by (auto intro!: integrable_bound[where g=f] integrable_abs)
lemma integrable_max:
assumes int: "integrable M f" "integrable M g"
shows "integrable M (\<lambda> x. max (f x) (g x))"
proof (rule integrable_bound)
show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
using int by (simp add: integrable_abs)
show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
using int unfolding integrable_def by auto
qed auto
lemma integrable_min:
assumes int: "integrable M f" "integrable M g"
shows "integrable M (\<lambda> x. min (f x) (g x))"
proof (rule integrable_bound)
show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
using int by (simp add: integrable_abs)
show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
using int unfolding integrable_def by auto
qed auto
lemma integral_triangle_inequality:
assumes "integrable M f"
shows "\<bar>integral\<^sup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
proof -
have "\<bar>integral\<^sup>L M f\<bar> = max (integral\<^sup>L M f) (- integral\<^sup>L M f)" by auto
also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
using assms integral_minus(2)[of M f, symmetric]
by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
finally show ?thesis .
qed
lemma integrable_nonneg:
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+ x. f x \<partial>M) \<noteq> \<infinity>"
shows "integrable M f"
unfolding integrable_def
proof (intro conjI f)
have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) = 0"
using f by (subst positive_integral_0_iff_AE) auto
then show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>" by simp
qed
lemma integral_positive:
assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
shows "0 \<le> integral\<^sup>L M f"
proof -
have "0 = (\<integral>x. 0 \<partial>M)" by auto
also have "\<dots> \<le> integral\<^sup>L M f"
using assms by (rule integral_mono[OF integral_zero(1)])
finally show ?thesis .
qed
lemma integral_monotone_convergence_pos:
assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
and pos: "\<And>i. AE x in M. 0 \<le> f i x"
and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
and u: "u \<in> borel_measurable M"
shows "integrable M u"
and "integral\<^sup>L M u = x"
proof -
have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^sup>+ x. ereal (f n x) \<partial>M))"
proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
fix i
from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
by eventually_elim (auto simp: mono_def)
show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
using i by auto
next
show "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = \<integral>\<^sup>+ x. (SUP i. ereal (f i x)) \<partial>M"
apply (rule positive_integral_cong_AE)
using lim mono
by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
qed
also have "\<dots> = ereal x"
using mono i unfolding positive_integral_eq_integral[OF i pos]
by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
finally have "(\<integral>\<^sup>+ x. ereal (u x) \<partial>M) = ereal x" .
moreover have "(\<integral>\<^sup>+ x. ereal (- u x) \<partial>M) = 0"
proof (subst positive_integral_0_iff_AE)
show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
using u by auto
from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
proof eventually_elim
fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
then show "ereal (- u x) \<le> 0"
using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
qed
qed
ultimately show "integrable M u" "integral\<^sup>L M u = x"
by (auto simp: integrable_def lebesgue_integral_def u)
qed
lemma integral_monotone_convergence:
assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
and ilim: "(\<lambda>i. integral\<^sup>L M (f i)) ----> x"
and u: "u \<in> borel_measurable M"
shows "integrable M u"
and "integral\<^sup>L M u = x"
proof -
have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
using f by auto
have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
using mono by (auto simp: mono_def le_fun_def)
have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
using mono by (auto simp: field_simps mono_def le_fun_def)
have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
using lim by (auto intro!: tendsto_diff)
have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^sup>L M (f 0)"
using f ilim by (auto intro!: tendsto_diff)
have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
using f[of 0] u by auto
note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6]
have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
using diff(1) f by (rule integral_add(1))
with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
by auto
qed
lemma integral_0_iff:
assumes "integrable M f"
shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
proof -
have *: "(\<integral>\<^sup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
"(\<integral>\<^sup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
from positive_integral_0_iff[OF this(1)] this(2)
show ?thesis unfolding lebesgue_integral_def *
using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
by (auto simp add: real_of_ereal_eq_0)
qed
lemma positive_integral_PInf:
assumes f: "f \<in> borel_measurable M"
and not_Inf: "integral\<^sup>P 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 positive_integral_cmult_indicator) (auto simp: measurable_sets)
also have "\<dots> \<le> integral\<^sup>P M (\<lambda>x. max 0 (f x))"
by (auto intro!: positive_integral_mono simp: indicator_def max_def)
finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^sup>P M f"
by (simp add: positive_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 positive_integral_PInf_AE:
assumes "f \<in> borel_measurable M" "integral\<^sup>P 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 positive_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 positive_integral_PInf)
show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
show "integral\<^sup>P M f \<noteq> \<infinity>"
using assms by (simp add: positive_integral_eq_simple_integral)
qed
lemma integral_real:
"AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^sup>P M f) - real (integral\<^sup>P M (\<lambda>x. - f x))"
using assms unfolding lebesgue_integral_def
by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
lemma (in finite_measure) lebesgue_integral_const[simp]:
shows "integrable M (\<lambda>x. a)"
and "(\<integral>x. a \<partial>M) = a * measure M (space M)"
proof -
{ fix a :: real assume "0 \<le> a"
then have "(\<integral>\<^sup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
by (subst positive_integral_const) auto
moreover
from `0 \<le> a` have "(\<integral>\<^sup>+ x. ereal (-a) \<partial>M) = 0"
by (subst positive_integral_0_iff_AE) auto
ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
note * = this
show "integrable M (\<lambda>x. a)"
proof cases
assume "0 \<le> a" with * show ?thesis .
next
assume "\<not> 0 \<le> a"
then have "0 \<le> -a" by auto
from *[OF this] show ?thesis by simp
qed
show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
qed
lemma (in finite_measure) integrable_const_bound:
assumes "AE x in M. \<bar>f x\<bar> \<le> B" and "f \<in> borel_measurable M" shows "integrable M f"
by (auto intro: integrable_bound[where f="\<lambda>x. B"] lebesgue_integral_const assms)
lemma indicator_less[simp]:
"indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
by (simp add: indicator_def not_le)
lemma (in finite_measure) integral_less_AE:
assumes int: "integrable M X" "integrable M Y"
assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
assumes gt: "AE x in M. X x \<le> Y x"
shows "integral\<^sup>L M X < integral\<^sup>L M Y"
proof -
have "integral\<^sup>L M X \<le> integral\<^sup>L M Y"
using gt int by (intro integral_mono_AE) auto
moreover
have "integral\<^sup>L M X \<noteq> integral\<^sup>L M Y"
proof
assume eq: "integral\<^sup>L M X = integral\<^sup>L M Y"
have "integral\<^sup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^sup>L M (\<lambda>x. Y x - X x)"
using gt by (intro integral_cong_AE) auto
also have "\<dots> = 0"
using eq int by simp
finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
using int by (simp add: integral_0_iff)
moreover
have "(\<integral>\<^sup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^sup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
using A by (intro positive_integral_mono_AE) auto
then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
using int A by (simp add: integrable_def)
ultimately have "emeasure M A = 0"
using emeasure_nonneg[of M A] by simp
with `(emeasure M) A \<noteq> 0` show False by auto
qed
ultimately show ?thesis by auto
qed
lemma (in finite_measure) integral_less_AE_space:
assumes int: "integrable M X" "integrable M Y"
assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
shows "integral\<^sup>L M X < integral\<^sup>L M Y"
using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
lemma integral_dominated_convergence:
assumes u[measurable]: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x"
and w[measurable]: "integrable M w"
and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
and [measurable]: "u' \<in> borel_measurable M"
shows "integrable M u'"
and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
and "(\<lambda>i. integral\<^sup>L M (u i)) ----> integral\<^sup>L M u'" (is ?lim)
proof -
have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x"
using bound by (auto simp: AE_all_countable)
with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x"
by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs)
from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x"
by eventually_elim auto
show "integrable M u'"
by (rule integrable_bound) fact+
let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>"
have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
using w u `integrable M u'` by (auto intro!: integrable_abs)
from u'_bound all_bound
have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x"
proof (eventually_elim, intro allI)
fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x"
then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
also have "\<dots> \<le> w x + w x"
using * by (intro add_mono) auto
finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp
qed
have PI_diff: "\<And>n. (\<integral>\<^sup>+ x. ereal (?diff n x) \<partial>M) =
(\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
using diff w diff_less_2w w_pos
by (subst positive_integral_diff[symmetric])
(auto simp: integrable_def intro!: positive_integral_cong_AE)
have "integrable M (\<lambda>x. 2 * w x)"
using w by auto
hence I2w_fin: "(\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
unfolding integrable_def by auto
have "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
proof cases
assume eq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
{ fix n
have "?f n \<le> ?wx" (is "integral\<^sup>P M ?f' \<le> _")
using diff_less_2w unfolding positive_integral_max_0
by (intro positive_integral_mono_AE) auto
then have "?f n = 0"
using positive_integral_positive[of M ?f'] eq_0 by auto }
then show ?thesis by (simp add: Limsup_const)
next
assume neq_0: "(\<integral>\<^sup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
by (simp add: Limsup_mono positive_integral_positive)
finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
have "?wx = (\<integral>\<^sup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
using u'
proof (intro positive_integral_cong_AE, eventually_elim)
fix x assume u': "(\<lambda>i. u i x) ----> u' x"
show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
unfolding ereal_max_0
proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
using u' by (safe intro!: tendsto_intros)
then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
by (auto intro!: tendsto_real_max)
qed (rule trivial_limit_sequentially)
qed
also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^sup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
using w u unfolding integrable_def
by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
also have "\<dots> = (\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M) -
limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
unfolding PI_diff positive_integral_max_0
using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
by (subst liminf_ereal_cminus) auto
finally show ?thesis
using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
unfolding positive_integral_max_0
by (cases rule: ereal2_cases[of "\<integral>\<^sup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
auto
qed
have "liminf ?f \<le> limsup ?f"
by (intro Liminf_le_Limsup trivial_limit_sequentially)
moreover
{ have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
also have "\<dots> \<le> liminf ?f"
by (simp add: Liminf_mono positive_integral_positive)
finally have "0 \<le> liminf ?f" . }
ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
using `limsup ?f = 0` by auto
have "\<And>n. (\<integral>\<^sup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
using diff positive_integral_positive[of M]
by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
then show ?lim_diff
using Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
by simp
show ?lim
proof (rule LIMSEQ_I)
fix r :: real assume "0 < r"
from LIMSEQ_D[OF `?lim_diff` this]
obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
using diff by (auto simp: integral_positive)
show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r"
proof (safe intro!: exI[of _ N])
fix n assume "N \<le> n"
have "\<bar>integral\<^sup>L M (u n) - integral\<^sup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
using u `integrable M u'` by auto
also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
by (rule_tac integral_triangle_inequality) auto
also note N[OF `N \<le> n`]
finally show "norm (integral\<^sup>L M (u n) - integral\<^sup>L M u') < r" by simp
qed
qed
qed
lemma integral_sums:
assumes integrable[measurable]: "\<And>i. integrable M (f i)"
and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
and "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
proof -
have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
using summable unfolding summable_def by auto
from bchoice[OF this]
obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
by (rule borel_measurable_LIMSEQ) auto
let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
using sums unfolding summable_def ..
have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
using integrable by auto
have 2: "\<And>j. AE x in M. \<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x"
using AE_space
proof eventually_elim
fix j x assume [simp]: "x \<in> space M"
have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
also have "\<dots> \<le> w x" using w[of x] series_pos_le[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
finally show "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp
qed
have 3: "integrable M ?w"
proof (rule integral_monotone_convergence(1))
let ?F = "\<lambda>n y. (\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
have "\<And>n. integrable M (?F n)"
using integrable by (auto intro!: integrable_abs)
thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
show "AE x in M. mono (\<lambda>n. ?w' n x)"
by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
using w by (simp_all add: tendsto_const sums_def)
have *: "\<And>n. integral\<^sup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
using integrable by (simp add: integrable_abs cong: integral_cong)
from abs_sum
show "(\<lambda>i. integral\<^sup>L M (?w' i)) ----> x" unfolding * sums_def .
qed (simp add: w_borel measurable_If_set)
from summable[THEN summable_rabs_cancel]
have 4: "AE x in M. (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
by (auto intro: summable_sumr_LIMSEQ_suminf)
note int = integral_dominated_convergence(1,3)[OF 1 2 3 4
borel_measurable_suminf[OF integrableD(1)[OF integrable]]]
from int show "integrable M ?S" by simp
show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF integrable]
using int(2) by simp
qed
lemma integrable_mult_indicator:
"A \<in> sets M \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. f x * indicator A x)"
by (rule integrable_bound[where f="\<lambda>x. \<bar>f x\<bar>"])
(auto intro: integrable_abs split: split_indicator)
lemma tendsto_integral_at_top:
fixes M :: "real measure"
assumes M: "sets M = sets borel" and f[measurable]: "integrable M f"
shows "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
proof -
have M_measure[simp]: "borel_measurable M = borel_measurable borel"
using M by (simp add: sets_eq_imp_space_eq measurable_def)
{ fix f assume f: "integrable M f" "\<And>x. 0 \<le> f x"
then have [measurable]: "f \<in> borel_measurable borel"
by (simp add: integrable_def)
have "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> \<integral> x. f x \<partial>M) at_top"
proof (rule tendsto_at_topI_sequentially)
have "\<And>j. AE x in M. \<bar>f x * indicator {.. j} x\<bar> \<le> f x"
using f(2) by (intro AE_I2) (auto split: split_indicator)
have int: "\<And>n. integrable M (\<lambda>x. f x * indicator {.. n} x)"
by (rule integrable_mult_indicator) (auto simp: M f)
show "(\<lambda>n. \<integral> x. f x * indicator {..real n} x \<partial>M) ----> integral\<^sup>L M f"
proof (rule integral_dominated_convergence)
{ fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
by (rule eventually_sequentiallyI[of "natceiling x"])
(auto split: split_indicator simp: natceiling_le_eq) }
from filterlim_cong[OF refl refl this]
show "AE x in M. (\<lambda>n. f x * indicator {..real n} x) ----> f x"
by (simp add: tendsto_const)
qed (fact+, simp)
show "mono (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
by (intro monoI integral_mono int) (auto split: split_indicator intro: f)
qed }
note nonneg = this
let ?P = "\<lambda>y. \<integral> x. max 0 (f x) * indicator {..y} x \<partial>M"
let ?N = "\<lambda>y. \<integral> x. max 0 (- f x) * indicator {..y} x \<partial>M"
let ?p = "integral\<^sup>L M (\<lambda>x. max 0 (f x))"
let ?n = "integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
have "(?P ---> ?p) at_top" "(?N ---> ?n) at_top"
by (auto intro!: nonneg integrable_max f)
note tendsto_diff[OF this]
also have "(\<lambda>y. ?P y - ?N y) = (\<lambda>y. \<integral> x. f x * indicator {..y} x \<partial>M)"
by (subst integral_diff(2)[symmetric])
(auto intro!: integrable_mult_indicator integrable_max f integral_cong ext
simp: M split: split_max)
also have "?p - ?n = integral\<^sup>L M f"
by (subst integral_diff(2)[symmetric])
(auto intro!: integrable_max f integral_cong ext simp: M split: split_max)
finally show ?thesis .
qed
lemma integral_monotone_convergence_at_top:
fixes M :: "real measure"
assumes M: "sets M = sets borel"
assumes nonneg: "AE x in M. 0 \<le> f x"
assumes borel: "f \<in> borel_measurable borel"
assumes int: "\<And>y. integrable M (\<lambda>x. f x * indicator {.. y} x)"
assumes conv: "((\<lambda>y. \<integral> x. f x * indicator {.. y} x \<partial>M) ---> x) at_top"
shows "integrable M f" "integral\<^sup>L M f = x"
proof -
from nonneg have "AE x in M. mono (\<lambda>n::nat. f x * indicator {..real n} x)"
by (auto split: split_indicator intro!: monoI)
{ fix x have "eventually (\<lambda>n. f x * indicator {..real n} x = f x) sequentially"
by (rule eventually_sequentiallyI[of "natceiling x"])
(auto split: split_indicator simp: natceiling_le_eq) }
from filterlim_cong[OF refl refl this]
have "AE x in M. (\<lambda>i. f x * indicator {..real i} x) ----> f x"
by (simp add: tendsto_const)
have "(\<lambda>i. \<integral> x. f x * indicator {..real i} x \<partial>M) ----> x"
using conv filterlim_real_sequentially by (rule filterlim_compose)
have M_measure[simp]: "borel_measurable M = borel_measurable borel"
using M by (simp add: sets_eq_imp_space_eq measurable_def)
have "f \<in> borel_measurable M"
using borel by simp
show "integrable M f"
by (rule integral_monotone_convergence) fact+
show "integral\<^sup>L M f = x"
by (rule integral_monotone_convergence) fact+
qed
section "Lebesgue integration on countable spaces"
lemma integral_on_countable:
assumes f: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
shows "integrable M f"
and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^sup>L M f" (is ?sums)
proof -
let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^sup>L M (?F r)"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
{ fix x assume "x \<in> space M"
hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
then obtain i where "i\<in>S" "enum i = f x" by auto
have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
proof cases
fix j assume "j = i"
thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
next
fix j assume "j \<noteq> i"
show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
qed
hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
have "(\<lambda>i. ?F i x) sums f x"
"(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
by (auto intro!: sums_single simp: F F_abs) }
note F_sums_f = this(1) and F_abs_sums_f = this(2)
have int_f: "integral\<^sup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
{ fix r
have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
by (auto simp: indicator_def intro!: integral_cong)
also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
note int_abs_F = this
have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
using F_abs_sums_f unfolding sums_iff by auto
from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
show ?sums unfolding enum_eq int_f by simp
from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
show "integrable M f" unfolding int_f by simp
qed
section {* Distributions *}
lemma positive_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>P (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 positive_integral_cong[of _ f g])
apply simp
apply (subst positive_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 positive_integral_cong[OF eq])
(auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
lemma positive_integral_distr:
"T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>P (distr M M' T) f = (\<integral>\<^sup>+ x. f (T x) \<partial>M)"
by (subst (1 2) positive_integral_max_0[symmetric])
(simp add: positive_integral_distr')
lemma integral_distr:
"T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^sup>L (distr M M' T) f = (\<integral> x. f (T x) \<partial>M)"
unfolding lebesgue_integral_def
by (subst (1 2) positive_integral_distr) auto
lemma integrable_distr_eq:
"T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))"
unfolding integrable_def
by (subst (1 2) positive_integral_distr) (auto simp: comp_def)
lemma integrable_distr:
"T \<in> measurable M M' \<Longrightarrow> integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
by (subst integrable_distr_eq[symmetric]) auto
section {* Lebesgue integration on @{const count_space} *}
lemma simple_function_count_space[simp]:
"simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
unfolding simple_function_def by simp
lemma positive_integral_count_space:
assumes A: "finite {a\<in>A. 0 < f a}"
shows "integral\<^sup>P (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!: positive_integral_cong
simp add: indicator_def if_distrib setsum_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 positive_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: positive_integral_cmult_indicator one_ereal_def[symmetric])
finally show ?thesis by (simp add: positive_integral_max_0)
qed
lemma integrable_count_space:
"finite X \<Longrightarrow> integrable (count_space X) f"
by (auto simp: positive_integral_count_space integrable_def)
lemma positive_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 positive_integral_max_0[symmetric])
(auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
lemma lebesgue_integral_count_space_finite_support:
assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
proof -
have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)"
"\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a"
"\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a"
"{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}"
"({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}"
by (auto split: split_max)
have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}"
by (auto intro: finite_subset[OF _ f])
then show ?thesis
unfolding lebesgue_integral_def
apply (subst (1 2) positive_integral_max_0[symmetric])
apply (subst (1 2) positive_integral_count_space)
apply (auto simp add: * setsum_negf setsum_Un
simp del: ereal_max)
done
qed
lemma lebesgue_integral_count_space_finite:
"finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
apply (auto intro!: setsum_mono_zero_left
simp: positive_integral_count_space_finite lebesgue_integral_def)
apply (subst (1 2) setsum_real_of_ereal[symmetric])
apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
done
lemma borel_measurable_count_space[simp, intro!]:
"f \<in> borel_measurable (count_space A)"
by simp
section {* Measure spaces with an associated density *}
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]: "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 positive_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: positive_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!: positive_integral_positive)
have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^sup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
apply (subst positive_integral_max_0[symmetric])
apply (intro ext positive_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: positive_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 positive_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 positive_integral_max_0[symmetric])
apply (intro positive_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 `A \<in> sets M`
by (intro positive_integral_0_iff) auto
also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
using f `A \<in> sets M`
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 positive_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>P (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 positive_integral_cong[OF cong(3)])
apply (simp_all cong: positive_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: positive_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: positive_integral_add ereal_zero_le_0_iff intro!: positive_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: SUPR_ereal_cmult seq)
from seq f show ?case
apply (simp add: positive_integral_monotone_convergence_SUP)
apply (subst positive_integral_cong_AE[OF eq])
apply (subst positive_integral_monotone_convergence_SUP_AE)
apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
done
qed
lemma positive_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>P (density M f) g' = (\<integral>\<^sup>+ x. f x * g' x \<partial>M)"
by (subst (1 2) positive_integral_max_0[symmetric])
(auto intro!: positive_integral_cong_AE
simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
lemma integral_density:
assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
and g: "g \<in> borel_measurable M"
shows "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
unfolding lebesgue_integral_def integrable_def using f g
by (auto simp: positive_integral_density)
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!: positive_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 default (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: positive_integral_cmult_indicator emeasure_density)
lemma measure_density_const:
"A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real 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 positive_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
positive_integral_distr measurable_sets cong: positive_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
section {* Point measure *}
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 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)
declare [[simproc del: finite_Collect]]
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 `X \<subseteq> A` by auto
with A show ?thesis
by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
point_measure_def indicator_def)
qed
declare [[simproc add: finite_Collect]]
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_zero_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_zero_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 positive_integral_point_measure:
"finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
integral\<^sup>P (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 positive_integral_density)
apply (simp_all add: AE_count_space positive_integral_density)
apply (subst positive_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 positive_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>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
lemma lebesgue_integral_point_measure_finite:
"finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^sup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
lemma integrable_point_measure_finite:
"finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
unfolding point_measure_def
apply (subst density_ereal_max_0)
apply (subst integral_density)
apply (auto simp: AE_count_space integrable_count_space)
done
section {* Uniform measure *}
definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
lemma
shows sets_uniform_measure[simp]: "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!: positive_integral_cong)
also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
using A B
by (subst positive_integral_cmult_indicator) (simp_all add: sets.Int emeasure_nonneg)
finally show ?thesis .
qed
lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
using emeasure_notin_sets[of A M] by blast
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)
section {* Uniform count measure *}
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 emeasure_uniform_count_measure:
"finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
by (simp add: real_eq_of_nat 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: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
end