(* Title: HOL/Probability/Bochner_Integration.thy
Author: Johannes Hölzl, TU München
*)
header {* Bochner Integration for Vector-Valued Functions *}
theory Bochner_Integration
imports Finite_Product_Measure
begin
text {*
In the following development of the Bochner integral we use second countable topologies instead
of separable spaces. A second countable topology is also separable.
*}
lemma borel_measurable_implies_sequence_metric:
fixes f :: "'a \<Rightarrow> 'b :: {metric_space, second_countable_topology}"
assumes [measurable]: "f \<in> borel_measurable M"
shows "\<exists>F. (\<forall>i. simple_function M (F i)) \<and> (\<forall>x\<in>space M. (\<lambda>i. F i x) ----> f x) \<and>
(\<forall>i. \<forall>x\<in>space M. dist (F i x) z \<le> 2 * dist (f x) z)"
proof -
obtain D :: "'b set" where "countable D" and D: "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
by (erule countable_dense_setE)
def e \<equiv> "from_nat_into D"
{ fix n x
obtain d where "d \<in> D" and d: "d \<in> ball x (1 / Suc n)"
using D[of "ball x (1 / Suc n)"] by auto
from `d \<in> D` D[of UNIV] `countable D` obtain i where "d = e i"
unfolding e_def by (auto dest: from_nat_into_surj)
with d have "\<exists>i. dist x (e i) < 1 / Suc n"
by auto }
note e = this
def A \<equiv> "\<lambda>m n. {x\<in>space M. dist (f x) (e n) < 1 / (Suc m) \<and> 1 / (Suc m) \<le> dist (f x) z}"
def B \<equiv> "\<lambda>m. disjointed (A m)"
def m \<equiv> "\<lambda>N x. Max {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
def F \<equiv> "\<lambda>N::nat. \<lambda>x. if (\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)) \<and> (\<exists>n\<le>N. x \<in> B (m N x) n)
then e (LEAST n. x \<in> B (m N x) n) else z"
have B_imp_A[intro, simp]: "\<And>x m n. x \<in> B m n \<Longrightarrow> x \<in> A m n"
using disjointed_subset[of "A m" for m] unfolding B_def by auto
{ fix m
have "\<And>n. A m n \<in> sets M"
by (auto simp: A_def)
then have "\<And>n. B m n \<in> sets M"
using sets.range_disjointed_sets[of "A m" M] by (auto simp: B_def) }
note this[measurable]
{ fix N i x assume "\<exists>m\<le>N. x \<in> (\<Union>n\<le>N. B m n)"
then have "m N x \<in> {m::nat. m \<le> N \<and> x \<in> (\<Union>n\<le>N. B m n)}"
unfolding m_def by (intro Max_in) auto
then have "m N x \<le> N" "\<exists>n\<le>N. x \<in> B (m N x) n"
by auto }
note m = this
{ fix j N i x assume "j \<le> N" "i \<le> N" "x \<in> B j i"
then have "j \<le> m N x"
unfolding m_def by (intro Max_ge) auto }
note m_upper = this
show ?thesis
unfolding simple_function_def
proof (safe intro!: exI[of _ F])
have [measurable]: "\<And>i. F i \<in> borel_measurable M"
unfolding F_def m_def by measurable
show "\<And>x i. F i -` {x} \<inter> space M \<in> sets M"
by measurable
{ fix i
{ fix n x assume "x \<in> B (m i x) n"
then have "(LEAST n. x \<in> B (m i x) n) \<le> n"
by (intro Least_le)
also assume "n \<le> i"
finally have "(LEAST n. x \<in> B (m i x) n) \<le> i" . }
then have "F i ` space M \<subseteq> {z} \<union> e ` {.. i}"
by (auto simp: F_def)
then show "finite (F i ` space M)"
by (rule finite_subset) auto }
{ fix N i n x assume "i \<le> N" "n \<le> N" "x \<in> B i n"
then have 1: "\<exists>m\<le>N. x \<in> (\<Union> n\<le>N. B m n)" by auto
from m[OF this] obtain n where n: "m N x \<le> N" "n \<le> N" "x \<in> B (m N x) n" by auto
moreover
def L \<equiv> "LEAST n. x \<in> B (m N x) n"
have "dist (f x) (e L) < 1 / Suc (m N x)"
proof -
have "x \<in> B (m N x) L"
using n(3) unfolding L_def by (rule LeastI)
then have "x \<in> A (m N x) L"
by auto
then show ?thesis
unfolding A_def by simp
qed
ultimately have "dist (f x) (F N x) < 1 / Suc (m N x)"
by (auto simp add: F_def L_def) }
note * = this
fix x assume "x \<in> space M"
show "(\<lambda>i. F i x) ----> f x"
proof cases
assume "f x = z"
then have "\<And>i n. x \<notin> A i n"
unfolding A_def by auto
then have "\<And>i. F i x = z"
by (auto simp: F_def)
then show ?thesis
using `f x = z` by auto
next
assume "f x \<noteq> z"
show ?thesis
proof (rule tendstoI)
fix e :: real assume "0 < e"
with `f x \<noteq> z` obtain n where "1 / Suc n < e" "1 / Suc n < dist (f x) z"
by (metis dist_nz order_less_trans neq_iff nat_approx_posE)
with `x\<in>space M` `f x \<noteq> z` have "x \<in> (\<Union>i. B n i)"
unfolding A_def B_def UN_disjointed_eq using e by auto
then obtain i where i: "x \<in> B n i" by auto
show "eventually (\<lambda>i. dist (F i x) (f x) < e) sequentially"
using eventually_ge_at_top[of "max n i"]
proof eventually_elim
fix j assume j: "max n i \<le> j"
with i have "dist (f x) (F j x) < 1 / Suc (m j x)"
by (intro *[OF _ _ i]) auto
also have "\<dots> \<le> 1 / Suc n"
using j m_upper[OF _ _ i]
by (auto simp: field_simps)
also note `1 / Suc n < e`
finally show "dist (F j x) (f x) < e"
by (simp add: less_imp_le dist_commute)
qed
qed
qed
fix i
{ fix n m assume "x \<in> A n m"
then have "dist (e m) (f x) + dist (f x) z \<le> 2 * dist (f x) z"
unfolding A_def by (auto simp: dist_commute)
also have "dist (e m) z \<le> dist (e m) (f x) + dist (f x) z"
by (rule dist_triangle)
finally (xtrans) have "dist (e m) z \<le> 2 * dist (f x) z" . }
then show "dist (F i x) z \<le> 2 * dist (f x) z"
unfolding F_def
apply auto
apply (rule LeastI2)
apply auto
done
qed
qed
lemma real_indicator: "real (indicator A x :: ereal) = indicator A x"
unfolding indicator_def by auto
lemma split_indicator_asm:
"P (indicator S x) \<longleftrightarrow> \<not> ((x \<in> S \<and> \<not> P 1) \<or> (x \<notin> S \<and> \<not> P 0))"
unfolding indicator_def by auto
lemma
fixes f :: "'a \<Rightarrow> 'b::semiring_1" assumes "finite A"
shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator (B x) (g x)) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator (B x) (g x) * f x) = (\<Sum>x\<in>{x\<in>A. g x \<in> B x}. f x)"
unfolding indicator_def
using assms by (auto intro!: setsum_mono_zero_cong_right split: split_if_asm)
lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
fixes P :: "('a \<Rightarrow> real) \<Rightarrow> bool"
assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
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> (\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow> (\<And>i x. 0 \<le> U i x) \<Longrightarrow> (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. U i x) ----> u x) \<Longrightarrow> P u"
shows "P u"
proof -
have "(\<lambda>x. ereal (u x)) \<in> borel_measurable M" using u by auto
from borel_measurable_implies_simple_function_sequence'[OF this]
obtain U where 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 (ereal (u x))" and nn: "\<And>i x. 0 \<le> U i x"
by blast
def U' \<equiv> "\<lambda>i x. indicator (space M) x * real (U i x)"
then have U'_sf[measurable]: "\<And>i. simple_function M (U' i)"
using U by (auto intro!: simple_function_compose1[where g=real])
show "P u"
proof (rule seq)
fix i show "U' i \<in> borel_measurable M" "\<And>x. 0 \<le> U' i x"
using U nn by (auto
intro: borel_measurable_simple_function
intro!: borel_measurable_real_of_ereal real_of_ereal_pos borel_measurable_times
simp: U'_def zero_le_mult_iff)
show "incseq U'"
using U(2,3) nn
by (auto simp: incseq_def le_fun_def image_iff eq_commute U'_def indicator_def
intro!: real_of_ereal_positive_mono)
next
fix x assume x: "x \<in> space M"
have "(\<lambda>i. U i x) ----> (SUP i. U i x)"
using U(2) by (intro LIMSEQ_SUP) (auto simp: incseq_def le_fun_def)
moreover have "(\<lambda>i. U i x) = (\<lambda>i. ereal (U' i x))"
using x nn U(3) by (auto simp: fun_eq_iff U'_def ereal_real image_iff eq_commute)
moreover have "(SUP i. U i x) = ereal (u x)"
using sup u(2) by (simp add: max_def)
ultimately show "(\<lambda>i. U' i x) ----> u x"
by simp
next
fix i
have "U' i ` space M \<subseteq> real ` (U i ` space M)" "finite (U i ` space M)"
unfolding U'_def using U(1) by (auto dest: simple_functionD)
then have fin: "finite (U' i ` space M)"
by (metis finite_subset finite_imageI)
moreover have "\<And>z. {y. U' i z = y \<and> y \<in> U' i ` space M \<and> z \<in> space M} = (if z \<in> space M then {U' i z} else {})"
by auto
ultimately have U': "(\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z) = U' i"
by (simp add: U'_def fun_eq_iff)
have "\<And>x. x \<in> U' i ` space M \<Longrightarrow> 0 \<le> x"
using nn by (auto simp: U'_def real_of_ereal_pos)
with fin have "P (\<lambda>z. \<Sum>y\<in>U' i`space M. y * indicator {x\<in>space M. U' i x = y} z)"
proof induct
case empty from set[of "{}"] show ?case
by (simp add: indicator_def[abs_def])
next
case (insert x F)
then show ?case
by (auto intro!: add mult set setsum_nonneg split: split_indicator split_indicator_asm
simp del: setsum_mult_indicator simp: setsum_nonneg_eq_0_iff )
qed
with U' show "P (U' i)" by simp
qed
qed
lemma scaleR_cong_right:
fixes x :: "'a :: real_vector"
shows "(x \<noteq> 0 \<Longrightarrow> r = p) \<Longrightarrow> r *\<^sub>R x = p *\<^sub>R x"
by (cases "x = 0") auto
inductive simple_bochner_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" for M f where
"simple_function M f \<Longrightarrow> emeasure M {y\<in>space M. f y \<noteq> 0} \<noteq> \<infinity> \<Longrightarrow>
simple_bochner_integrable M f"
lemma simple_bochner_integrable_compose2:
assumes p_0: "p 0 0 = 0"
shows "simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integrable M g \<Longrightarrow>
simple_bochner_integrable M (\<lambda>x. p (f x) (g x))"
proof (safe intro!: simple_bochner_integrable.intros elim!: simple_bochner_integrable.cases del: notI)
assume sf: "simple_function M f" "simple_function M g"
then show "simple_function M (\<lambda>x. p (f x) (g x))"
by (rule simple_function_compose2)
from sf have [measurable]:
"f \<in> measurable M (count_space UNIV)"
"g \<in> measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function)
assume fin: "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" "emeasure M {y \<in> space M. g y \<noteq> 0} \<noteq> \<infinity>"
have "emeasure M {x\<in>space M. p (f x) (g x) \<noteq> 0} \<le>
emeasure M ({x\<in>space M. f x \<noteq> 0} \<union> {x\<in>space M. g x \<noteq> 0})"
by (intro emeasure_mono) (auto simp: p_0)
also have "\<dots> \<le> emeasure M {x\<in>space M. f x \<noteq> 0} + emeasure M {x\<in>space M. g x \<noteq> 0}"
by (intro emeasure_subadditive) auto
finally show "emeasure M {y \<in> space M. p (f y) (g y) \<noteq> 0} \<noteq> \<infinity>"
using fin by auto
qed
lemma simple_function_finite_support:
assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>" and nn: "\<And>x. 0 \<le> f x"
shows "emeasure M {x\<in>space M. f x \<noteq> 0} \<noteq> \<infinity>"
proof cases
from f have meas[measurable]: "f \<in> borel_measurable M"
by (rule borel_measurable_simple_function)
assume non_empty: "\<exists>x\<in>space M. f x \<noteq> 0"
def m \<equiv> "Min (f`space M - {0})"
have "m \<in> f`space M - {0}"
unfolding m_def using f non_empty by (intro Min_in) (auto simp: simple_function_def)
then have m: "0 < m"
using nn by (auto simp: less_le)
from m have "m * emeasure M {x\<in>space M. 0 \<noteq> f x} =
(\<integral>\<^sup>+x. m * indicator {x\<in>space M. 0 \<noteq> f x} x \<partial>M)"
using f by (intro nn_integral_cmult_indicator[symmetric]) auto
also have "\<dots> \<le> (\<integral>\<^sup>+x. f x \<partial>M)"
using AE_space
proof (intro nn_integral_mono_AE, eventually_elim)
fix x assume "x \<in> space M"
with nn show "m * indicator {x \<in> space M. 0 \<noteq> f x} x \<le> f x"
using f by (auto split: split_indicator simp: simple_function_def m_def)
qed
also note `\<dots> < \<infinity>`
finally show ?thesis
using m by auto
next
assume "\<not> (\<exists>x\<in>space M. f x \<noteq> 0)"
with nn have *: "{x\<in>space M. f x \<noteq> 0} = {}"
by auto
show ?thesis unfolding * by simp
qed
lemma simple_bochner_integrableI_bounded:
assumes f: "simple_function M f" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
shows "simple_bochner_integrable M f"
proof
have "emeasure M {y \<in> space M. ereal (norm (f y)) \<noteq> 0} \<noteq> \<infinity>"
proof (rule simple_function_finite_support)
show "simple_function M (\<lambda>x. ereal (norm (f x)))"
using f by (rule simple_function_compose1)
show "(\<integral>\<^sup>+ y. ereal (norm (f y)) \<partial>M) < \<infinity>" by fact
qed simp
then show "emeasure M {y \<in> space M. f y \<noteq> 0} \<noteq> \<infinity>" by simp
qed fact
definition simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
"simple_bochner_integral M f = (\<Sum>y\<in>f`space M. measure M {x\<in>space M. f x = y} *\<^sub>R y)"
lemma simple_bochner_integral_partition:
assumes f: "simple_bochner_integrable M f" and g: "simple_function M g"
assumes sub: "\<And>x y. x \<in> space M \<Longrightarrow> y \<in> space M \<Longrightarrow> g x = g y \<Longrightarrow> f x = f y"
assumes v: "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)"
shows "simple_bochner_integral M f = (\<Sum>y\<in>g ` space M. measure M {x\<in>space M. g x = y} *\<^sub>R v y)"
(is "_ = ?r")
proof -
from f g have [simp]: "finite (f`space M)" "finite (g`space M)"
by (auto simp: simple_function_def elim: simple_bochner_integrable.cases)
from f have [measurable]: "f \<in> measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
from g have [measurable]: "g \<in> measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
{ fix y assume "y \<in> space M"
then have "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
by (auto cong: sub simp: v[symmetric]) }
note eq = this
have "simple_bochner_integral M f =
(\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} else 0) *\<^sub>R y)"
unfolding simple_bochner_integral_def
proof (safe intro!: setsum_cong scaleR_cong_right)
fix y assume y: "y \<in> space M" "f y \<noteq> 0"
have [simp]: "g ` space M \<inter> {z. \<exists>x\<in>space M. f y = f x \<and> z = g x} =
{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
by auto
have eq:"{x \<in> space M. f x = f y} =
(\<Union>i\<in>{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}. {x \<in> space M. g x = i})"
by (auto simp: eq_commute cong: sub rev_conj_cong)
have "finite (g`space M)" by simp
then have "finite {z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
by (rule rev_finite_subset) auto
moreover
{ fix x assume "x \<in> space M" "f x = f y"
then have "x \<in> space M" "f x \<noteq> 0"
using y by auto
then have "emeasure M {y \<in> space M. g y = g x} \<le> emeasure M {y \<in> space M. f y \<noteq> 0}"
by (auto intro!: emeasure_mono cong: sub)
then have "emeasure M {xa \<in> space M. g xa = g x} < \<infinity>"
using f by (auto simp: simple_bochner_integrable.simps) }
ultimately
show "measure M {x \<in> space M. f x = f y} =
(\<Sum>z\<in>g ` space M. if \<exists>x\<in>space M. f y = f x \<and> z = g x then measure M {x \<in> space M. g x = z} else 0)"
apply (simp add: setsum_cases eq)
apply (subst measure_finite_Union[symmetric])
apply (auto simp: disjoint_family_on_def)
done
qed
also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
if \<exists>x\<in>space M. y = f x \<and> z = g x then measure M {x\<in>space M. g x = z} *\<^sub>R y else 0))"
by (auto intro!: setsum_cong simp: scaleR_setsum_left)
also have "\<dots> = ?r"
by (subst setsum_commute)
(auto intro!: setsum_cong simp: setsum_cases scaleR_setsum_right[symmetric] eq)
finally show "simple_bochner_integral M f = ?r" .
qed
lemma simple_bochner_integral_add:
assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
shows "simple_bochner_integral M (\<lambda>x. f x + g x) =
simple_bochner_integral M f + simple_bochner_integral M g"
proof -
from f g have "simple_bochner_integral M (\<lambda>x. f x + g x) =
(\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R (fst y + snd y))"
by (intro simple_bochner_integral_partition)
(auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
moreover from f g have "simple_bochner_integral M f =
(\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R fst y)"
by (intro simple_bochner_integral_partition)
(auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
moreover from f g have "simple_bochner_integral M g =
(\<Sum>y\<in>(\<lambda>x. (f x, g x)) ` space M. measure M {x \<in> space M. (f x, g x) = y} *\<^sub>R snd y)"
by (intro simple_bochner_integral_partition)
(auto simp: simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
ultimately show ?thesis
by (simp add: setsum_addf[symmetric] scaleR_add_right)
qed
lemma (in linear) simple_bochner_integral_linear:
assumes g: "simple_bochner_integrable M g"
shows "simple_bochner_integral M (\<lambda>x. f (g x)) = f (simple_bochner_integral M g)"
proof -
from g have "simple_bochner_integral M (\<lambda>x. f (g x)) =
(\<Sum>y\<in>g ` space M. measure M {x \<in> space M. g x = y} *\<^sub>R f y)"
by (intro simple_bochner_integral_partition)
(auto simp: simple_bochner_integrable_compose2[where p="\<lambda>x y. f x"] zero
elim: simple_bochner_integrable.cases)
also have "\<dots> = f (simple_bochner_integral M g)"
by (simp add: simple_bochner_integral_def setsum scaleR)
finally show ?thesis .
qed
lemma simple_bochner_integral_minus:
assumes f: "simple_bochner_integrable M f"
shows "simple_bochner_integral M (\<lambda>x. - f x) = - simple_bochner_integral M f"
proof -
interpret linear uminus by unfold_locales auto
from f show ?thesis
by (rule simple_bochner_integral_linear)
qed
lemma simple_bochner_integral_diff:
assumes f: "simple_bochner_integrable M f" and g: "simple_bochner_integrable M g"
shows "simple_bochner_integral M (\<lambda>x. f x - g x) =
simple_bochner_integral M f - simple_bochner_integral M g"
unfolding diff_conv_add_uminus using f g
by (subst simple_bochner_integral_add)
(auto simp: simple_bochner_integral_minus simple_bochner_integrable_compose2[where p="\<lambda>x y. - y"])
lemma simple_bochner_integral_norm_bound:
assumes f: "simple_bochner_integrable M f"
shows "norm (simple_bochner_integral M f) \<le> simple_bochner_integral M (\<lambda>x. norm (f x))"
proof -
have "norm (simple_bochner_integral M f) \<le>
(\<Sum>y\<in>f ` space M. norm (measure M {x \<in> space M. f x = y} *\<^sub>R y))"
unfolding simple_bochner_integral_def by (rule norm_setsum)
also have "\<dots> = (\<Sum>y\<in>f ` space M. measure M {x \<in> space M. f x = y} *\<^sub>R norm y)"
by (simp add: measure_nonneg)
also have "\<dots> = simple_bochner_integral M (\<lambda>x. norm (f x))"
using f
by (intro simple_bochner_integral_partition[symmetric])
(auto intro: f simple_bochner_integrable_compose2 elim: simple_bochner_integrable.cases)
finally show ?thesis .
qed
lemma simple_bochner_integral_eq_nn_integral:
assumes f: "simple_bochner_integrable M f" "\<And>x. 0 \<le> f x"
shows "simple_bochner_integral M f = (\<integral>\<^sup>+x. f x \<partial>M)"
proof -
{ fix x y z have "(x \<noteq> 0 \<Longrightarrow> y = z) \<Longrightarrow> ereal x * y = ereal x * z"
by (cases "x = 0") (auto simp: zero_ereal_def[symmetric]) }
note ereal_cong_mult = this
have [measurable]: "f \<in> borel_measurable M"
using f(1) by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
{ fix y assume y: "y \<in> space M" "f y \<noteq> 0"
have "ereal (measure M {x \<in> space M. f x = f y}) = emeasure M {x \<in> space M. f x = f y}"
proof (rule emeasure_eq_ereal_measure[symmetric])
have "emeasure M {x \<in> space M. f x = f y} \<le> emeasure M {x \<in> space M. f x \<noteq> 0}"
using y by (intro emeasure_mono) auto
with f show "emeasure M {x \<in> space M. f x = f y} \<noteq> \<infinity>"
by (auto simp: simple_bochner_integrable.simps)
qed
moreover have "{x \<in> space M. f x = f y} = (\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M"
by auto
ultimately have "ereal (measure M {x \<in> space M. f x = f y}) =
emeasure M ((\<lambda>x. ereal (f x)) -` {ereal (f y)} \<inter> space M)" by simp }
with f have "simple_bochner_integral M f = (\<integral>\<^sup>Sx. f x \<partial>M)"
unfolding simple_integral_def
by (subst simple_bochner_integral_partition[OF f(1), where g="\<lambda>x. ereal (f x)" and v=real])
(auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
intro!: setsum_cong ereal_cong_mult
simp: setsum_ereal[symmetric] times_ereal.simps(1)[symmetric] mult_ac
simp del: setsum_ereal times_ereal.simps(1))
also have "\<dots> = (\<integral>\<^sup>+x. f x \<partial>M)"
using f
by (intro nn_integral_eq_simple_integral[symmetric])
(auto simp: simple_function_compose1 simple_bochner_integrable.simps)
finally show ?thesis .
qed
lemma simple_bochner_integral_bounded:
fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector, second_countable_topology}"
assumes f[measurable]: "f \<in> borel_measurable M"
assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
shows "ereal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \<le>
(\<integral>\<^sup>+ x. norm (f x - s x) \<partial>M) + (\<integral>\<^sup>+ x. norm (f x - t x) \<partial>M)"
(is "ereal (norm (?s - ?t)) \<le> ?S + ?T")
proof -
have [measurable]: "s \<in> borel_measurable M" "t \<in> borel_measurable M"
using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
have "ereal (norm (?s - ?t)) = norm (simple_bochner_integral M (\<lambda>x. s x - t x))"
using s t by (subst simple_bochner_integral_diff) auto
also have "\<dots> \<le> simple_bochner_integral M (\<lambda>x. norm (s x - t x))"
using simple_bochner_integrable_compose2[of "op -" M "s" "t"] s t
by (auto intro!: simple_bochner_integral_norm_bound)
also have "\<dots> = (\<integral>\<^sup>+x. norm (s x - t x) \<partial>M)"
using simple_bochner_integrable_compose2[of "\<lambda>x y. norm (x - y)" M "s" "t"] s t
by (auto intro!: simple_bochner_integral_eq_nn_integral)
also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s x)) + ereal (norm (f x - t x)) \<partial>M)"
by (auto intro!: nn_integral_mono)
(metis (erased, hide_lams) add_diff_cancel_left add_diff_eq diff_add_eq order_trans
norm_minus_commute norm_triangle_ineq4 order_refl)
also have "\<dots> = ?S + ?T"
by (rule nn_integral_add) auto
finally show ?thesis .
qed
inductive has_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b::{real_normed_vector, second_countable_topology} \<Rightarrow> bool"
for M f x where
"f \<in> borel_measurable M \<Longrightarrow>
(\<And>i. simple_bochner_integrable M (s i)) \<Longrightarrow>
(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0 \<Longrightarrow>
(\<lambda>i. simple_bochner_integral M (s i)) ----> x \<Longrightarrow>
has_bochner_integral M f x"
lemma has_bochner_integral_cong:
assumes "M = N" "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" "x = y"
shows "has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral N g y"
unfolding has_bochner_integral.simps assms(1,3)
using assms(2) by (simp cong: measurable_cong_strong nn_integral_cong_strong)
lemma has_bochner_integral_cong_AE:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow>
has_bochner_integral M f x \<longleftrightarrow> has_bochner_integral M g x"
unfolding has_bochner_integral.simps
by (intro arg_cong[where f=Ex] ext conj_cong rev_conj_cong refl arg_cong[where f="\<lambda>x. x ----> 0"]
nn_integral_cong_AE)
auto
lemma borel_measurable_has_bochner_integral[measurable_dest]:
"has_bochner_integral M f x \<Longrightarrow> f \<in> borel_measurable M"
by (auto elim: has_bochner_integral.cases)
lemma has_bochner_integral_simple_bochner_integrable:
"simple_bochner_integrable M f \<Longrightarrow> has_bochner_integral M f (simple_bochner_integral M f)"
by (rule has_bochner_integral.intros[where s="\<lambda>_. f"])
(auto intro: borel_measurable_simple_function
elim: simple_bochner_integrable.cases
simp: zero_ereal_def[symmetric])
lemma has_bochner_integral_real_indicator:
assumes [measurable]: "A \<in> sets M" and A: "emeasure M A < \<infinity>"
shows "has_bochner_integral M (indicator A) (measure M A)"
proof -
have sbi: "simple_bochner_integrable M (indicator A::'a \<Rightarrow> real)"
proof
have "{y \<in> space M. (indicator A y::real) \<noteq> 0} = A"
using sets.sets_into_space[OF `A\<in>sets M`] by (auto split: split_indicator)
then show "emeasure M {y \<in> space M. (indicator A y::real) \<noteq> 0} \<noteq> \<infinity>"
using A by auto
qed (rule simple_function_indicator assms)+
moreover have "simple_bochner_integral M (indicator A) = measure M A"
using simple_bochner_integral_eq_nn_integral[OF sbi] A
by (simp add: ereal_indicator emeasure_eq_ereal_measure)
ultimately show ?thesis
by (metis has_bochner_integral_simple_bochner_integrable)
qed
lemma has_bochner_integral_add[intro]:
"has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
fix sf sg
assume f_sf: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - sf i x) \<partial>M) ----> 0"
assume g_sg: "(\<lambda>i. \<integral>\<^sup>+ x. norm (g x - sg i x) \<partial>M) ----> 0"
assume sf: "\<forall>i. simple_bochner_integrable M (sf i)"
and sg: "\<forall>i. simple_bochner_integrable M (sg i)"
then have [measurable]: "\<And>i. sf i \<in> borel_measurable M" "\<And>i. sg i \<in> borel_measurable M"
by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
assume [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
show "\<And>i. simple_bochner_integrable M (\<lambda>x. sf i x + sg i x)"
using sf sg by (simp add: simple_bochner_integrable_compose2)
show "(\<lambda>i. \<integral>\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \<partial>M) ----> 0"
(is "?f ----> 0")
proof (rule tendsto_sandwich)
show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
by (auto simp: nn_integral_nonneg)
show "eventually (\<lambda>i. ?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) \<partial>M) + \<integral>\<^sup>+ x. (norm (g x - sg i x)) \<partial>M) sequentially"
(is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
proof (intro always_eventually allI)
fix i have "?f i \<le> (\<integral>\<^sup>+ x. (norm (f x - sf i x)) + ereal (norm (g x - sg i x)) \<partial>M)"
by (auto intro!: nn_integral_mono norm_diff_triangle_ineq)
also have "\<dots> = ?g i"
by (intro nn_integral_add) auto
finally show "?f i \<le> ?g i" .
qed
show "?g ----> 0"
using tendsto_add_ereal[OF _ _ f_sf g_sg] by simp
qed
qed (auto simp: simple_bochner_integral_add tendsto_add)
lemma has_bochner_integral_bounded_linear:
assumes "bounded_linear T"
shows "has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M (\<lambda>x. T (f x)) (T x)"
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
interpret T: bounded_linear T by fact
have [measurable]: "T \<in> borel_measurable borel"
by (intro borel_measurable_continuous_on1 T.continuous_on continuous_on_id)
assume [measurable]: "f \<in> borel_measurable M"
then show "(\<lambda>x. T (f x)) \<in> borel_measurable M"
by auto
fix s assume f_s: "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0"
assume s: "\<forall>i. simple_bochner_integrable M (s i)"
then show "\<And>i. simple_bochner_integrable M (\<lambda>x. T (s i x))"
by (auto intro: simple_bochner_integrable_compose2 T.zero)
have [measurable]: "\<And>i. s i \<in> borel_measurable M"
using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
obtain K where K: "K > 0" "\<And>x i. norm (T (f x) - T (s i x)) \<le> norm (f x - s i x) * K"
using T.pos_bounded by (auto simp: T.diff[symmetric])
show "(\<lambda>i. \<integral>\<^sup>+ x. norm (T (f x) - T (s i x)) \<partial>M) ----> 0"
(is "?f ----> 0")
proof (rule tendsto_sandwich)
show "eventually (\<lambda>n. 0 \<le> ?f n) sequentially" "(\<lambda>_. 0) ----> 0"
by (auto simp: nn_integral_nonneg)
show "eventually (\<lambda>i. ?f i \<le> K * (\<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M)) sequentially"
(is "eventually (\<lambda>i. ?f i \<le> ?g i) sequentially")
proof (intro always_eventually allI)
fix i have "?f i \<le> (\<integral>\<^sup>+ x. ereal K * norm (f x - s i x) \<partial>M)"
using K by (intro nn_integral_mono) (auto simp: mult_ac)
also have "\<dots> = ?g i"
using K by (intro nn_integral_cmult) auto
finally show "?f i \<le> ?g i" .
qed
show "?g ----> 0"
using ereal_lim_mult[OF f_s, of "ereal K"] by simp
qed
assume "(\<lambda>i. simple_bochner_integral M (s i)) ----> x"
with s show "(\<lambda>i. simple_bochner_integral M (\<lambda>x. T (s i x))) ----> T x"
by (auto intro!: T.tendsto simp: T.simple_bochner_integral_linear)
qed
lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (\<lambda>x. 0) 0"
by (auto intro!: has_bochner_integral.intros[where s="\<lambda>_ _. 0"]
simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps
simple_bochner_integral_def image_constant_conv)
lemma has_bochner_integral_scaleR_left[intro]:
"(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x *\<^sub>R c) (x *\<^sub>R c)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_left])
lemma has_bochner_integral_scaleR_right[intro]:
"(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c *\<^sub>R f x) (c *\<^sub>R x)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_scaleR_right])
lemma has_bochner_integral_mult_left[intro]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_left])
lemma has_bochner_integral_mult_right[intro]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c * f x) (c * x)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_mult_right])
lemmas has_bochner_integral_divide =
has_bochner_integral_bounded_linear[OF bounded_linear_divide]
lemma has_bochner_integral_divide_zero[intro]:
fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x / c) (x / c)"
using has_bochner_integral_divide by (cases "c = 0") auto
lemma has_bochner_integral_inner_left[intro]:
"(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. f x \<bullet> c) (x \<bullet> c)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_left])
lemma has_bochner_integral_inner_right[intro]:
"(c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x) \<Longrightarrow> has_bochner_integral M (\<lambda>x. c \<bullet> f x) (c \<bullet> x)"
by (cases "c = 0") (auto simp add: has_bochner_integral_bounded_linear[OF bounded_linear_inner_right])
lemmas has_bochner_integral_minus =
has_bochner_integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
lemmas has_bochner_integral_Re =
has_bochner_integral_bounded_linear[OF bounded_linear_Re]
lemmas has_bochner_integral_Im =
has_bochner_integral_bounded_linear[OF bounded_linear_Im]
lemmas has_bochner_integral_cnj =
has_bochner_integral_bounded_linear[OF bounded_linear_cnj]
lemmas has_bochner_integral_of_real =
has_bochner_integral_bounded_linear[OF bounded_linear_of_real]
lemmas has_bochner_integral_fst =
has_bochner_integral_bounded_linear[OF bounded_linear_fst]
lemmas has_bochner_integral_snd =
has_bochner_integral_bounded_linear[OF bounded_linear_snd]
lemma has_bochner_integral_indicator:
"A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
has_bochner_integral M (\<lambda>x. indicator A x *\<^sub>R c) (measure M A *\<^sub>R c)"
by (intro has_bochner_integral_scaleR_left has_bochner_integral_real_indicator)
lemma has_bochner_integral_diff:
"has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M g y \<Longrightarrow>
has_bochner_integral M (\<lambda>x. f x - g x) (x - y)"
unfolding diff_conv_add_uminus
by (intro has_bochner_integral_add has_bochner_integral_minus)
lemma has_bochner_integral_setsum:
"(\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)) \<Longrightarrow>
has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (\<Sum>i\<in>I. x i)"
by (induct I rule: infinite_finite_induct) auto
lemma has_bochner_integral_implies_finite_norm:
"has_bochner_integral M f x \<Longrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
proof (elim has_bochner_integral.cases)
fix s v
assume [measurable]: "f \<in> borel_measurable M" and s: "\<And>i. simple_bochner_integrable M (s i)" and
lim_0: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
from order_tendstoD[OF lim_0, of "\<infinity>"]
obtain i where f_s_fin: "(\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) < \<infinity>"
by (metis (mono_tags, lifting) eventually_False_sequentially eventually_elim1
less_ereal.simps(4) zero_ereal_def)
have [measurable]: "\<And>i. s i \<in> borel_measurable M"
using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
def m \<equiv> "if space M = {} then 0 else Max ((\<lambda>x. norm (s i x))`space M)"
have "finite (s i ` space M)"
using s by (auto simp: simple_function_def simple_bochner_integrable.simps)
then have "finite (norm ` s i ` space M)"
by (rule finite_imageI)
then have "\<And>x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> m" "0 \<le> m"
by (auto simp: m_def image_comp comp_def Max_ge_iff)
then have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. ereal m * indicator {x\<in>space M. s i x \<noteq> 0} x \<partial>M)"
by (auto split: split_indicator intro!: Max_ge nn_integral_mono simp:)
also have "\<dots> < \<infinity>"
using s by (subst nn_integral_cmult_indicator) (auto simp: `0 \<le> m` simple_bochner_integrable.simps)
finally have s_fin: "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>" .
have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x - s i x)) + ereal (norm (s i x)) \<partial>M)"
by (auto intro!: nn_integral_mono) (metis add_commute norm_triangle_sub)
also have "\<dots> = (\<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) + (\<integral>\<^sup>+x. norm (s i x) \<partial>M)"
by (rule nn_integral_add) auto
also have "\<dots> < \<infinity>"
using s_fin f_s_fin by auto
finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
qed
lemma has_bochner_integral_norm_bound:
assumes i: "has_bochner_integral M f x"
shows "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
using assms proof
fix s assume
x: "(\<lambda>i. simple_bochner_integral M (s i)) ----> x" (is "?s ----> x") and
s[simp]: "\<And>i. simple_bochner_integrable M (s i)" and
lim: "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0" and
f[measurable]: "f \<in> borel_measurable M"
have [measurable]: "\<And>i. s i \<in> borel_measurable M"
using s by (auto simp: simple_bochner_integrable.simps intro: borel_measurable_simple_function)
show "norm x \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
proof (rule LIMSEQ_le)
show "(\<lambda>i. ereal (norm (?s i))) ----> norm x"
using x by (intro tendsto_intros lim_ereal[THEN iffD2])
show "\<exists>N. \<forall>n\<ge>N. norm (?s n) \<le> (\<integral>\<^sup>+x. norm (f x - s n x) \<partial>M) + (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
(is "\<exists>N. \<forall>n\<ge>N. _ \<le> ?t n")
proof (intro exI allI impI)
fix n
have "ereal (norm (?s n)) \<le> simple_bochner_integral M (\<lambda>x. norm (s n x))"
by (auto intro!: simple_bochner_integral_norm_bound)
also have "\<dots> = (\<integral>\<^sup>+x. norm (s n x) \<partial>M)"
by (intro simple_bochner_integral_eq_nn_integral)
(auto intro: s simple_bochner_integrable_compose2)
also have "\<dots> \<le> (\<integral>\<^sup>+x. ereal (norm (f x - s n x)) + norm (f x) \<partial>M)"
by (auto intro!: nn_integral_mono)
(metis add_commute norm_minus_commute norm_triangle_sub)
also have "\<dots> = ?t n"
by (rule nn_integral_add) auto
finally show "norm (?s n) \<le> ?t n" .
qed
have "?t ----> 0 + (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
using has_bochner_integral_implies_finite_norm[OF i]
by (intro tendsto_add_ereal tendsto_const lim) auto
then show "?t ----> \<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M"
by simp
qed
qed
lemma has_bochner_integral_eq:
"has_bochner_integral M f x \<Longrightarrow> has_bochner_integral M f y \<Longrightarrow> x = y"
proof (elim has_bochner_integral.cases)
assume f[measurable]: "f \<in> borel_measurable M"
fix s t
assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
assume "(\<lambda>i. \<integral>\<^sup>+ x. norm (f x - t i x) \<partial>M) ----> 0" (is "?T ----> 0")
assume s: "\<And>i. simple_bochner_integrable M (s i)"
assume t: "\<And>i. simple_bochner_integrable M (t i)"
have [measurable]: "\<And>i. s i \<in> borel_measurable M" "\<And>i. t i \<in> borel_measurable M"
using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
let ?s = "\<lambda>i. simple_bochner_integral M (s i)"
let ?t = "\<lambda>i. simple_bochner_integral M (t i)"
assume "?s ----> x" "?t ----> y"
then have "(\<lambda>i. norm (?s i - ?t i)) ----> norm (x - y)"
by (intro tendsto_intros)
moreover
have "(\<lambda>i. ereal (norm (?s i - ?t i))) ----> ereal 0"
proof (rule tendsto_sandwich)
show "eventually (\<lambda>i. 0 \<le> ereal (norm (?s i - ?t i))) sequentially" "(\<lambda>_. 0) ----> ereal 0"
by (auto simp: nn_integral_nonneg zero_ereal_def[symmetric])
show "eventually (\<lambda>i. norm (?s i - ?t i) \<le> ?S i + ?T i) sequentially"
by (intro always_eventually allI simple_bochner_integral_bounded s t f)
show "(\<lambda>i. ?S i + ?T i) ----> ereal 0"
using tendsto_add_ereal[OF _ _ `?S ----> 0` `?T ----> 0`]
by (simp add: zero_ereal_def[symmetric])
qed
then have "(\<lambda>i. norm (?s i - ?t i)) ----> 0"
by simp
ultimately have "norm (x - y) = 0"
by (rule LIMSEQ_unique)
then show "x = y" by simp
qed
lemma has_bochner_integralI_AE:
assumes f: "has_bochner_integral M f x"
and g: "g \<in> borel_measurable M"
and ae: "AE x in M. f x = g x"
shows "has_bochner_integral M g x"
using f
proof (safe intro!: has_bochner_integral.intros elim!: has_bochner_integral.cases)
fix s assume "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
also have "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) = (\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M)"
using ae
by (intro ext nn_integral_cong_AE, eventually_elim) simp
finally show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (g x - s i x)) \<partial>M) ----> 0" .
qed (auto intro: g)
lemma has_bochner_integral_eq_AE:
assumes f: "has_bochner_integral M f x"
and g: "has_bochner_integral M g y"
and ae: "AE x in M. f x = g x"
shows "x = y"
proof -
from assms have "has_bochner_integral M g x"
by (auto intro: has_bochner_integralI_AE)
from this g show "x = y"
by (rule has_bochner_integral_eq)
qed
lemma simple_bochner_integrable_restrict_space:
fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
shows "simple_bochner_integrable (restrict_space M \<Omega>) f \<longleftrightarrow>
simple_bochner_integrable M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
by (simp add: simple_bochner_integrable.simps space_restrict_space
simple_function_restrict_space[OF \<Omega>] emeasure_restrict_space[OF \<Omega>] Collect_restrict
indicator_eq_0_iff conj_ac)
lemma simple_bochner_integral_restrict_space:
fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
assumes f: "simple_bochner_integrable (restrict_space M \<Omega>) f"
shows "simple_bochner_integral (restrict_space M \<Omega>) f =
simple_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
proof -
have "finite ((\<lambda>x. indicator \<Omega> x *\<^sub>R f x)`space M)"
using f simple_bochner_integrable_restrict_space[OF \<Omega>, of f]
by (simp add: simple_bochner_integrable.simps simple_function_def)
then show ?thesis
by (auto simp: space_restrict_space measure_restrict_space[OF \<Omega>(1)] le_infI2
simple_bochner_integral_def Collect_restrict
split: split_indicator split_indicator_asm
intro!: setsum_mono_zero_cong_left arg_cong2[where f=measure])
qed
inductive integrable for M f where
"has_bochner_integral M f x \<Longrightarrow> integrable M f"
definition lebesgue_integral ("integral\<^sup>L") where
"integral\<^sup>L M f = (THE x. has_bochner_integral M f x)"
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 (\<lambda>x. f)"
lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x \<Longrightarrow> integral\<^sup>L M f = x"
by (metis the_equality has_bochner_integral_eq lebesgue_integral_def)
lemma has_bochner_integral_integrable:
"integrable M f \<Longrightarrow> has_bochner_integral M f (integral\<^sup>L M f)"
by (auto simp: has_bochner_integral_integral_eq integrable.simps)
lemma has_bochner_integral_iff:
"has_bochner_integral M f x \<longleftrightarrow> integrable M f \<and> integral\<^sup>L M f = x"
by (metis has_bochner_integral_integrable has_bochner_integral_integral_eq integrable.intros)
lemma simple_bochner_integrable_eq_integral:
"simple_bochner_integrable M f \<Longrightarrow> simple_bochner_integral M f = integral\<^sup>L M f"
using has_bochner_integral_simple_bochner_integrable[of M f]
by (simp add: has_bochner_integral_integral_eq)
lemma not_integrable_integral_eq: "\<not> integrable M f \<Longrightarrow> integral\<^sup>L M f = (THE x. False)"
unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])
lemma integral_eq_cases:
"integrable M f \<longleftrightarrow> integrable N g \<Longrightarrow>
(integrable M f \<Longrightarrow> integrable N g \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g) \<Longrightarrow>
integral\<^sup>L M f = integral\<^sup>L N g"
by (metis not_integrable_integral_eq)
lemma borel_measurable_integrable[measurable_dest]: "integrable M f \<Longrightarrow> f \<in> borel_measurable M"
by (auto elim: integrable.cases has_bochner_integral.cases)
lemma integrable_cong:
"M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable N g"
using assms by (simp cong: has_bochner_integral_cong add: integrable.simps)
lemma integrable_cong_AE:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
integrable M f \<longleftrightarrow> integrable M g"
unfolding integrable.simps
by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
lemma integral_cong:
"M = N \<Longrightarrow> (\<And>x. x \<in> space N \<Longrightarrow> f x = g x) \<Longrightarrow> integral\<^sup>L M f = integral\<^sup>L N g"
using assms by (simp cong: has_bochner_integral_cong add: lebesgue_integral_def)
lemma integral_cong_AE:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. f x = g x \<Longrightarrow>
integral\<^sup>L M f = integral\<^sup>L M g"
unfolding lebesgue_integral_def
by (intro has_bochner_integral_cong_AE arg_cong[where f=The] ext)
lemma integrable_add[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x + g x)"
by (auto simp: integrable.simps)
lemma integrable_zero[simp, intro]: "integrable M (\<lambda>x. 0)"
by (metis has_bochner_integral_zero integrable.simps)
lemma integrable_setsum[simp, intro]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow> integrable M (\<lambda>x. \<Sum>i\<in>I. f i x)"
by (metis has_bochner_integral_setsum integrable.simps)
lemma integrable_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
integrable M (\<lambda>x. indicator A x *\<^sub>R c)"
by (metis has_bochner_integral_indicator integrable.simps)
lemma integrable_real_indicator[simp, intro]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
integrable M (indicator A :: 'a \<Rightarrow> real)"
by (metis has_bochner_integral_real_indicator integrable.simps)
lemma integrable_diff[simp, intro]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> integrable M (\<lambda>x. f x - g x)"
by (auto simp: integrable.simps intro: has_bochner_integral_diff)
lemma integrable_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow> integrable M (\<lambda>x. T (f x))"
by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)
lemma integrable_scaleR_left[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x *\<^sub>R c)"
unfolding integrable.simps by fastforce
lemma integrable_scaleR_right[simp, intro]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c *\<^sub>R f x)"
unfolding integrable.simps by fastforce
lemma integrable_mult_left[simp, intro]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x * c)"
unfolding integrable.simps by fastforce
lemma integrable_mult_right[simp, intro]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c * f x)"
unfolding integrable.simps by fastforce
lemma integrable_divide_zero[simp, intro]:
fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x / c)"
unfolding integrable.simps by fastforce
lemma integrable_inner_left[simp, intro]:
"(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. f x \<bullet> c)"
unfolding integrable.simps by fastforce
lemma integrable_inner_right[simp, intro]:
"(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integrable M (\<lambda>x. c \<bullet> f x)"
unfolding integrable.simps by fastforce
lemmas integrable_minus[simp, intro] =
integrable_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
lemmas integrable_divide[simp, intro] =
integrable_bounded_linear[OF bounded_linear_divide]
lemmas integrable_Re[simp, intro] =
integrable_bounded_linear[OF bounded_linear_Re]
lemmas integrable_Im[simp, intro] =
integrable_bounded_linear[OF bounded_linear_Im]
lemmas integrable_cnj[simp, intro] =
integrable_bounded_linear[OF bounded_linear_cnj]
lemmas integrable_of_real[simp, intro] =
integrable_bounded_linear[OF bounded_linear_of_real]
lemmas integrable_fst[simp, intro] =
integrable_bounded_linear[OF bounded_linear_fst]
lemmas integrable_snd[simp, intro] =
integrable_bounded_linear[OF bounded_linear_snd]
lemma integral_zero[simp]: "integral\<^sup>L M (\<lambda>x. 0) = 0"
by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)
lemma integral_add[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
integral\<^sup>L M (\<lambda>x. f x + g x) = integral\<^sup>L M f + integral\<^sup>L M g"
by (intro has_bochner_integral_integral_eq has_bochner_integral_add has_bochner_integral_integrable)
lemma integral_diff[simp]: "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow>
integral\<^sup>L M (\<lambda>x. f x - g x) = integral\<^sup>L M f - integral\<^sup>L M g"
by (intro has_bochner_integral_integral_eq has_bochner_integral_diff has_bochner_integral_integrable)
lemma integral_setsum[simp]: "(\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)) \<Longrightarrow>
integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
by (intro has_bochner_integral_integral_eq has_bochner_integral_setsum has_bochner_integral_integrable)
lemma integral_bounded_linear: "bounded_linear T \<Longrightarrow> integrable M f \<Longrightarrow>
integral\<^sup>L M (\<lambda>x. T (f x)) = T (integral\<^sup>L M f)"
by (metis has_bochner_integral_bounded_linear has_bochner_integral_integrable has_bochner_integral_integral_eq)
lemma integral_indicator[simp]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
integral\<^sup>L M (\<lambda>x. indicator A x *\<^sub>R c) = measure M A *\<^sub>R c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_indicator has_bochner_integral_integrable)
lemma integral_real_indicator[simp]: "A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow>
integral\<^sup>L M (indicator A :: 'a \<Rightarrow> real) = measure M A"
by (intro has_bochner_integral_integral_eq has_bochner_integral_real_indicator has_bochner_integral_integrable)
lemma integral_scaleR_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x *\<^sub>R c \<partial>M) = integral\<^sup>L M f *\<^sub>R c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_left)
lemma integral_scaleR_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c *\<^sub>R f x \<partial>M) = c *\<^sub>R integral\<^sup>L M f"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_scaleR_right)
lemma integral_mult_left[simp]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x * c \<partial>M) = integral\<^sup>L M f * c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_left)
lemma integral_mult_right[simp]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c * f x \<partial>M) = c * integral\<^sup>L M f"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_mult_right)
lemma integral_inner_left[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. f x \<bullet> c \<partial>M) = integral\<^sup>L M f \<bullet> c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)
lemma integral_inner_right[simp]: "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> (\<integral> x. c \<bullet> f x \<partial>M) = c \<bullet> integral\<^sup>L M f"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_right)
lemma integral_divide_zero[simp]:
fixes c :: "_::{real_normed_field, field_inverse_zero, second_countable_topology}"
shows "(c \<noteq> 0 \<Longrightarrow> integrable M f) \<Longrightarrow> integral\<^sup>L M (\<lambda>x. f x / c) = integral\<^sup>L M f / c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_divide_zero)
lemmas integral_minus[simp] =
integral_bounded_linear[OF bounded_linear_minus[OF bounded_linear_ident]]
lemmas integral_divide[simp] =
integral_bounded_linear[OF bounded_linear_divide]
lemmas integral_Re[simp] =
integral_bounded_linear[OF bounded_linear_Re]
lemmas integral_Im[simp] =
integral_bounded_linear[OF bounded_linear_Im]
lemmas integral_cnj[simp] =
integral_bounded_linear[OF bounded_linear_cnj]
lemmas integral_of_real[simp] =
integral_bounded_linear[OF bounded_linear_of_real]
lemmas integral_fst[simp] =
integral_bounded_linear[OF bounded_linear_fst]
lemmas integral_snd[simp] =
integral_bounded_linear[OF bounded_linear_snd]
lemma integral_norm_bound_ereal:
"integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>\<^sup>+x. norm (f x) \<partial>M)"
by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)
lemma integrableI_sequence:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes f[measurable]: "f \<in> borel_measurable M"
assumes s: "\<And>i. simple_bochner_integrable M (s i)"
assumes lim: "(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) ----> 0" (is "?S ----> 0")
shows "integrable M f"
proof -
let ?s = "\<lambda>n. simple_bochner_integral M (s n)"
have "\<exists>x. ?s ----> x"
unfolding convergent_eq_cauchy
proof (rule metric_CauchyI)
fix e :: real assume "0 < e"
then have "0 < ereal (e / 2)" by auto
from order_tendstoD(2)[OF lim this]
obtain M where M: "\<And>n. M \<le> n \<Longrightarrow> ?S n < e / 2"
by (auto simp: eventually_sequentially)
show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (?s m) (?s n) < e"
proof (intro exI allI impI)
fix m n assume m: "M \<le> m" and n: "M \<le> n"
have "?S n \<noteq> \<infinity>"
using M[OF n] by auto
have "norm (?s n - ?s m) \<le> ?S n + ?S m"
by (intro simple_bochner_integral_bounded s f)
also have "\<dots> < ereal (e / 2) + e / 2"
using ereal_add_strict_mono[OF less_imp_le[OF M[OF n]] _ `?S n \<noteq> \<infinity>` M[OF m]]
by (auto simp: nn_integral_nonneg)
also have "\<dots> = e" by simp
finally show "dist (?s n) (?s m) < e"
by (simp add: dist_norm)
qed
qed
then obtain x where "?s ----> x" ..
show ?thesis
by (rule, rule) fact+
qed
lemma nn_integral_dominated_convergence_norm:
fixes u' :: "_ \<Rightarrow> _::{real_normed_vector, second_countable_topology}"
assumes [measurable]:
"\<And>i. u i \<in> borel_measurable M" "u' \<in> borel_measurable M" "w \<in> borel_measurable M"
and bound: "\<And>j. AE x in M. norm (u j x) \<le> w x"
and w: "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
shows "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> 0"
proof -
have "AE x in M. \<forall>j. norm (u j x) \<le> w x"
unfolding AE_all_countable by rule fact
with u' have bnd: "AE x in M. \<forall>j. norm (u' x - u j x) \<le> 2 * w x"
proof (eventually_elim, intro allI)
fix i x assume "(\<lambda>i. u i x) ----> u' x" "\<forall>j. norm (u j x) \<le> w x" "\<forall>j. norm (u j x) \<le> w x"
then have "norm (u' x) \<le> w x" "norm (u i x) \<le> w x"
by (auto intro: LIMSEQ_le_const2 tendsto_norm)
then have "norm (u' x) + norm (u i x) \<le> 2 * w x"
by simp
also have "norm (u' x - u i x) \<le> norm (u' x) + norm (u i x)"
by (rule norm_triangle_ineq4)
finally (xtrans) show "norm (u' x - u i x) \<le> 2 * w x" .
qed
have "(\<lambda>i. (\<integral>\<^sup>+x. norm (u' x - u i x) \<partial>M)) ----> (\<integral>\<^sup>+x. 0 \<partial>M)"
proof (rule nn_integral_dominated_convergence)
show "(\<integral>\<^sup>+x. 2 * w x \<partial>M) < \<infinity>"
by (rule nn_integral_mult_bounded_inf[OF _ w, of 2]) auto
show "AE x in M. (\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
using u'
proof eventually_elim
fix x assume "(\<lambda>i. u i x) ----> u' x"
from tendsto_diff[OF tendsto_const[of "u' x"] this]
show "(\<lambda>i. ereal (norm (u' x - u i x))) ----> 0"
by (simp add: zero_ereal_def tendsto_norm_zero_iff)
qed
qed (insert bnd, auto)
then show ?thesis by simp
qed
lemma integrableI_bounded:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes f[measurable]: "f \<in> borel_measurable M" and fin: "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
shows "integrable M f"
proof -
from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
s: "\<And>i. simple_function M (s i)" and
pointwise: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x" and
bound: "\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
by (simp add: norm_conv_dist) metis
show ?thesis
proof (rule integrableI_sequence)
{ fix i
have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
by (intro nn_integral_mono) (simp add: bound)
also have "\<dots> = 2 * (\<integral>\<^sup>+x. ereal (norm (f x)) \<partial>M)"
by (rule nn_integral_cmult) auto
finally have "(\<integral>\<^sup>+x. norm (s i x) \<partial>M) < \<infinity>"
using fin by auto }
note fin_s = this
show "\<And>i. simple_bochner_integrable M (s i)"
by (rule simple_bochner_integrableI_bounded) fact+
show "(\<lambda>i. \<integral>\<^sup>+ x. ereal (norm (f x - s i x)) \<partial>M) ----> 0"
proof (rule nn_integral_dominated_convergence_norm)
show "\<And>j. AE x in M. norm (s j x) \<le> 2 * norm (f x)"
using bound by auto
show "\<And>i. s i \<in> borel_measurable M" "(\<lambda>x. 2 * norm (f x)) \<in> borel_measurable M"
using s by (auto intro: borel_measurable_simple_function)
show "(\<integral>\<^sup>+ x. ereal (2 * norm (f x)) \<partial>M) < \<infinity>"
using fin unfolding times_ereal.simps(1)[symmetric] by (subst nn_integral_cmult) auto
show "AE x in M. (\<lambda>i. s i x) ----> f x"
using pointwise by auto
qed fact
qed fact
qed
lemma integrableI_nonneg:
fixes f :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>M) < \<infinity>"
shows "integrable M f"
proof -
have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>\<^sup>+x. f x \<partial>M)"
using assms by (intro nn_integral_cong_AE) auto
then show ?thesis
using assms by (intro integrableI_bounded) auto
qed
lemma integrable_iff_bounded:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and> (\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
using integrableI_bounded[of f M] has_bochner_integral_implies_finite_norm[of M f]
unfolding integrable.simps has_bochner_integral.simps[abs_def] by auto
lemma integrable_bound:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
and g :: "'a \<Rightarrow> 'c::{banach, second_countable_topology}"
shows "integrable M f \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. norm (g x) \<le> norm (f x)) \<Longrightarrow>
integrable M g"
unfolding integrable_iff_bounded
proof safe
assume "f \<in> borel_measurable M" "g \<in> borel_measurable M"
assume "AE x in M. norm (g x) \<le> norm (f x)"
then have "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
by (intro nn_integral_mono_AE) auto
also assume "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
finally show "(\<integral>\<^sup>+ x. ereal (norm (g x)) \<partial>M) < \<infinity>" .
qed
lemma integrable_abs[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
using assms by (rule integrable_bound) auto
lemma integrable_norm[simp, intro]:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes [measurable]: "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
using assms by (rule integrable_bound) auto
lemma integrable_norm_cancel:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes [measurable]: "integrable M (\<lambda>x. norm (f x))" "f \<in> borel_measurable M" shows "integrable M f"
using assms by (rule integrable_bound) auto
lemma integrable_abs_cancel:
fixes f :: "'a \<Rightarrow> real"
assumes [measurable]: "integrable M (\<lambda>x. \<bar>f x\<bar>)" "f \<in> borel_measurable M" shows "integrable M f"
using assms by (rule integrable_bound) auto
lemma integrable_max[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes fg[measurable]: "integrable M f" "integrable M g"
shows "integrable M (\<lambda>x. max (f x) (g x))"
using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
by (rule integrable_bound) auto
lemma integrable_min[simp, intro]:
fixes f :: "'a \<Rightarrow> real"
assumes fg[measurable]: "integrable M f" "integrable M g"
shows "integrable M (\<lambda>x. min (f x) (g x))"
using integrable_add[OF integrable_norm[OF fg(1)] integrable_norm[OF fg(2)]]
by (rule integrable_bound) auto
lemma integral_minus_iff[simp]:
"integrable M (\<lambda>x. - f x ::'a::{banach, second_countable_topology}) \<longleftrightarrow> integrable M f"
unfolding integrable_iff_bounded
by (auto intro: borel_measurable_uminus[of "\<lambda>x. - f x" M, simplified])
lemma integrable_indicator_iff:
"integrable M (indicator A::_ \<Rightarrow> real) \<longleftrightarrow> A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>"
by (simp add: integrable_iff_bounded borel_measurable_indicator_iff ereal_indicator nn_integral_indicator'
cong: conj_cong)
lemma
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and w :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M" "integrable M w"
assumes lim: "AE x in M. (\<lambda>i. s i x) ----> f x"
assumes bound: "\<And>i. AE x in M. norm (s i x) \<le> w x"
shows integrable_dominated_convergence: "integrable M f"
and integrable_dominated_convergence2: "\<And>i. integrable M (s i)"
and integral_dominated_convergence: "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f"
proof -
have "AE x in M. 0 \<le> w x"
using bound[of 0] by eventually_elim (auto intro: norm_ge_zero order_trans)
then have "(\<integral>\<^sup>+x. w x \<partial>M) = (\<integral>\<^sup>+x. norm (w x) \<partial>M)"
by (intro nn_integral_cong_AE) auto
with `integrable M w` have w: "w \<in> borel_measurable M" "(\<integral>\<^sup>+x. w x \<partial>M) < \<infinity>"
unfolding integrable_iff_bounded by auto
show int_s: "\<And>i. integrable M (s i)"
unfolding integrable_iff_bounded
proof
fix i
have "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
using bound by (intro nn_integral_mono_AE) auto
with w show "(\<integral>\<^sup>+ x. ereal (norm (s i x)) \<partial>M) < \<infinity>" by auto
qed fact
have all_bound: "AE x in M. \<forall>i. norm (s i x) \<le> w x"
using bound unfolding AE_all_countable by auto
show int_f: "integrable M f"
unfolding integrable_iff_bounded
proof
have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) \<le> (\<integral>\<^sup>+x. w x \<partial>M)"
using all_bound lim
proof (intro nn_integral_mono_AE, eventually_elim)
fix x assume "\<forall>i. norm (s i x) \<le> w x" "(\<lambda>i. s i x) ----> f x"
then show "ereal (norm (f x)) \<le> ereal (w x)"
by (intro LIMSEQ_le_const2[where X="\<lambda>i. ereal (norm (s i x))"] tendsto_intros lim_ereal[THEN iffD2]) auto
qed
with w show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" by auto
qed fact
have "(\<lambda>n. ereal (norm (integral\<^sup>L M (s n) - integral\<^sup>L M f))) ----> ereal 0" (is "?d ----> ereal 0")
proof (rule tendsto_sandwich)
show "eventually (\<lambda>n. ereal 0 \<le> ?d n) sequentially" "(\<lambda>_. ereal 0) ----> ereal 0" by auto
show "eventually (\<lambda>n. ?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)) sequentially"
proof (intro always_eventually allI)
fix n
have "?d n = norm (integral\<^sup>L M (\<lambda>x. s n x - f x))"
using int_f int_s by simp
also have "\<dots> \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)"
by (intro int_f int_s integrable_diff integral_norm_bound_ereal)
finally show "?d n \<le> (\<integral>\<^sup>+x. norm (s n x - f x) \<partial>M)" .
qed
show "(\<lambda>n. \<integral>\<^sup>+x. norm (s n x - f x) \<partial>M) ----> ereal 0"
unfolding zero_ereal_def[symmetric]
apply (subst norm_minus_commute)
proof (rule nn_integral_dominated_convergence_norm[where w=w])
show "\<And>n. s n \<in> borel_measurable M"
using int_s unfolding integrable_iff_bounded by auto
qed fact+
qed
then have "(\<lambda>n. integral\<^sup>L M (s n) - integral\<^sup>L M f) ----> 0"
unfolding lim_ereal tendsto_norm_zero_iff .
from tendsto_add[OF this tendsto_const[of "integral\<^sup>L M f"]]
show "(\<lambda>i. integral\<^sup>L M (s i)) ----> integral\<^sup>L M f" by simp
qed
lemma integrable_mult_left_iff:
fixes f :: "'a \<Rightarrow> real"
shows "integrable M (\<lambda>x. c * f x) \<longleftrightarrow> c = 0 \<or> integrable M f"
using integrable_mult_left[of c M f] integrable_mult_left[of "1 / c" M "\<lambda>x. c * f x"]
by (cases "c = 0") auto
lemma nn_integral_eq_integral:
assumes f: "integrable M f"
assumes nonneg: "AE x in M. 0 \<le> f x"
shows "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
proof -
{ fix f :: "'a \<Rightarrow> real" assume f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" "integrable M f"
then have "(\<integral>\<^sup>+ x. f x \<partial>M) = integral\<^sup>L M f"
proof (induct rule: borel_measurable_induct_real)
case (set A) then show ?case
by (simp add: integrable_indicator_iff ereal_indicator emeasure_eq_ereal_measure)
next
case (mult f c) then show ?case
unfolding times_ereal.simps(1)[symmetric]
by (subst nn_integral_cmult)
(auto simp add: integrable_mult_left_iff zero_ereal_def[symmetric])
next
case (add g f)
then have "integrable M f" "integrable M g"
by (auto intro!: integrable_bound[OF add(8)])
with add show ?case
unfolding plus_ereal.simps(1)[symmetric]
by (subst nn_integral_add) auto
next
case (seq s)
{ fix i x assume "x \<in> space M" with seq(4) have "s i x \<le> f x"
by (intro LIMSEQ_le_const[OF seq(5)] exI[of _ i]) (auto simp: incseq_def le_fun_def) }
note s_le_f = this
show ?case
proof (rule LIMSEQ_unique)
show "(\<lambda>i. ereal (integral\<^sup>L M (s i))) ----> ereal (integral\<^sup>L M f)"
unfolding lim_ereal
proof (rule integral_dominated_convergence[where w=f])
show "integrable M f" by fact
from s_le_f seq show "\<And>i. AE x in M. norm (s i x) \<le> f x"
by auto
qed (insert seq, auto)
have int_s: "\<And>i. integrable M (s i)"
using seq f s_le_f by (intro integrable_bound[OF f(3)]) auto
have "(\<lambda>i. \<integral>\<^sup>+ x. s i x \<partial>M) ----> \<integral>\<^sup>+ x. f x \<partial>M"
using seq s_le_f f
by (intro nn_integral_dominated_convergence[where w=f])
(auto simp: integrable_iff_bounded)
also have "(\<lambda>i. \<integral>\<^sup>+x. s i x \<partial>M) = (\<lambda>i. \<integral>x. s i x \<partial>M)"
using seq int_s by simp
finally show "(\<lambda>i. \<integral>x. s i x \<partial>M) ----> \<integral>\<^sup>+x. f x \<partial>M"
by simp
qed
qed }
from this[of "\<lambda>x. max 0 (f x)"] assms have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = integral\<^sup>L M (\<lambda>x. max 0 (f x))"
by simp
also have "\<dots> = integral\<^sup>L M f"
using assms by (auto intro!: integral_cong_AE)
also have "(\<integral>\<^sup>+ x. max 0 (f x) \<partial>M) = (\<integral>\<^sup>+ x. f x \<partial>M)"
using assms by (auto intro!: nn_integral_cong_AE simp: max_def)
finally show ?thesis .
qed
lemma
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes integrable[measurable]: "\<And>i. integrable M (f i)"
and summable: "AE x in M. summable (\<lambda>i. norm (f i x))"
and sums: "summable (\<lambda>i. (\<integral>x. norm (f i x) \<partial>M))"
shows integrable_suminf: "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
and sums_integral: "(\<lambda>i. integral\<^sup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is "?f sums ?x")
and integral_suminf: "(\<integral>x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^sup>L M (f i))"
and summable_integral: "summable (\<lambda>i. integral\<^sup>L M (f i))"
proof -
have 1: "integrable M (\<lambda>x. \<Sum>i. norm (f i x))"
proof (rule integrableI_bounded)
have "(\<integral>\<^sup>+ x. ereal (norm (\<Sum>i. norm (f i x))) \<partial>M) = (\<integral>\<^sup>+ x. (\<Sum>i. ereal (norm (f i x))) \<partial>M)"
apply (intro nn_integral_cong_AE)
using summable
apply eventually_elim
apply (simp add: suminf_ereal' suminf_nonneg)
done
also have "\<dots> = (\<Sum>i. \<integral>\<^sup>+ x. norm (f i x) \<partial>M)"
by (intro nn_integral_suminf) auto
also have "\<dots> = (\<Sum>i. ereal (\<integral>x. norm (f i x) \<partial>M))"
by (intro arg_cong[where f=suminf] ext nn_integral_eq_integral integrable_norm integrable) auto
finally show "(\<integral>\<^sup>+ x. ereal (norm (\<Sum>i. norm (f i x))) \<partial>M) < \<infinity>"
by (simp add: suminf_ereal' sums)
qed simp
have 2: "AE x in M. (\<lambda>n. \<Sum>i<n. f i x) ----> (\<Sum>i. f i x)"
using summable by eventually_elim (auto intro: summable_LIMSEQ summable_norm_cancel)
have 3: "\<And>j. AE x in M. norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))"
using summable
proof eventually_elim
fix j x assume [simp]: "summable (\<lambda>i. norm (f i x))"
have "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i<j. norm (f i x))" by (rule norm_setsum)
also have "\<dots> \<le> (\<Sum>i. norm (f i x))"
using setsum_le_suminf[of "\<lambda>i. norm (f i x)"] unfolding sums_iff by auto
finally show "norm (\<Sum>i<j. f i x) \<le> (\<Sum>i. norm (f i x))" by simp
qed
note ibl = integrable_dominated_convergence[OF _ _ 1 2 3]
note int = integral_dominated_convergence[OF _ _ 1 2 3]
show "integrable M ?S"
by (rule ibl) measurable
show "?f sums ?x" unfolding sums_def
using int by (simp add: integrable)
then show "?x = suminf ?f" "summable ?f"
unfolding sums_iff by auto
qed
lemma integral_norm_bound:
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
shows "integrable M f \<Longrightarrow> norm (integral\<^sup>L M f) \<le> (\<integral>x. norm (f x) \<partial>M)"
using nn_integral_eq_integral[of M "\<lambda>x. norm (f x)"]
using integral_norm_bound_ereal[of M f] by simp
lemma integral_eq_nn_integral:
"integrable M f \<Longrightarrow> (\<And>x. 0 \<le> f x) \<Longrightarrow> integral\<^sup>L M f = real (\<integral>\<^sup>+ x. ereal (f x) \<partial>M)"
by (subst nn_integral_eq_integral) auto
lemma integrableI_simple_bochner_integrable:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "simple_bochner_integrable M f \<Longrightarrow> integrable M f"
by (intro integrableI_sequence[where s="\<lambda>_. f"] borel_measurable_simple_function)
(auto simp: zero_ereal_def[symmetric] simple_bochner_integrable.simps)
lemma integrable_induct[consumes 1, case_names base add lim, induct pred: integrable]:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "integrable M f"
assumes base: "\<And>A c. A \<in> sets M \<Longrightarrow> emeasure M A < \<infinity> \<Longrightarrow> P (\<lambda>x. indicator A x *\<^sub>R c)"
assumes add: "\<And>f g. integrable M f \<Longrightarrow> P f \<Longrightarrow> integrable M g \<Longrightarrow> P g \<Longrightarrow> P (\<lambda>x. f x + g x)"
assumes lim: "\<And>f s. (\<And>i. integrable M (s i)) \<Longrightarrow> (\<And>i. P (s i)) \<Longrightarrow>
(\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x) \<Longrightarrow>
(\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)) \<Longrightarrow> integrable M f \<Longrightarrow> P f"
shows "P f"
proof -
from `integrable M f` have f: "f \<in> borel_measurable M" "(\<integral>\<^sup>+x. norm (f x) \<partial>M) < \<infinity>"
unfolding integrable_iff_bounded by auto
from borel_measurable_implies_sequence_metric[OF f(1)]
obtain s where s: "\<And>i. simple_function M (s i)" "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. s i x) ----> f x"
"\<And>i x. x \<in> space M \<Longrightarrow> norm (s i x) \<le> 2 * norm (f x)"
unfolding norm_conv_dist by metis
{ fix f A
have [simp]: "P (\<lambda>x. 0)"
using base[of "{}" undefined] by simp
have "(\<And>i::'b. i \<in> A \<Longrightarrow> integrable M (f i::'a \<Rightarrow> 'b)) \<Longrightarrow>
(\<And>i. i \<in> A \<Longrightarrow> P (f i)) \<Longrightarrow> P (\<lambda>x. \<Sum>i\<in>A. f i x)"
by (induct A rule: infinite_finite_induct) (auto intro!: add) }
note setsum = this
def s' \<equiv> "\<lambda>i z. indicator (space M) z *\<^sub>R s i z"
then have s'_eq_s: "\<And>i x. x \<in> space M \<Longrightarrow> s' i x = s i x"
by simp
have sf[measurable]: "\<And>i. simple_function M (s' i)"
unfolding s'_def using s(1)
by (intro simple_function_compose2[where h="op *\<^sub>R"] simple_function_indicator) auto
{ fix i
have "\<And>z. {y. s' i z = y \<and> y \<in> s' i ` space M \<and> y \<noteq> 0 \<and> z \<in> space M} =
(if z \<in> space M \<and> s' i z \<noteq> 0 then {s' i z} else {})"
by (auto simp add: s'_def split: split_indicator)
then have "\<And>z. s' i = (\<lambda>z. \<Sum>y\<in>s' i`space M - {0}. indicator {x\<in>space M. s' i x = y} z *\<^sub>R y)"
using sf by (auto simp: fun_eq_iff simple_function_def s'_def) }
note s'_eq = this
show "P f"
proof (rule lim)
fix i
have "(\<integral>\<^sup>+x. norm (s' i x) \<partial>M) \<le> (\<integral>\<^sup>+x. 2 * ereal (norm (f x)) \<partial>M)"
using s by (intro nn_integral_mono) (auto simp: s'_eq_s)
also have "\<dots> < \<infinity>"
using f by (subst nn_integral_cmult) auto
finally have sbi: "simple_bochner_integrable M (s' i)"
using sf by (intro simple_bochner_integrableI_bounded) auto
then show "integrable M (s' i)"
by (rule integrableI_simple_bochner_integrable)
{ fix x assume"x \<in> space M" "s' i x \<noteq> 0"
then have "emeasure M {y \<in> space M. s' i y = s' i x} \<le> emeasure M {y \<in> space M. s' i y \<noteq> 0}"
by (intro emeasure_mono) auto
also have "\<dots> < \<infinity>"
using sbi by (auto elim: simple_bochner_integrable.cases)
finally have "emeasure M {y \<in> space M. s' i y = s' i x} \<noteq> \<infinity>" by simp }
then show "P (s' i)"
by (subst s'_eq) (auto intro!: setsum base)
fix x assume "x \<in> space M" with s show "(\<lambda>i. s' i x) ----> f x"
by (simp add: s'_eq_s)
show "norm (s' i x) \<le> 2 * norm (f x)"
using `x \<in> space M` s by (simp add: s'_eq_s)
qed fact
qed
lemma integral_nonneg_AE:
fixes f :: "'a \<Rightarrow> real"
assumes [measurable]: "integrable M f" "AE x in M. 0 \<le> f x"
shows "0 \<le> integral\<^sup>L M f"
proof -
have "0 \<le> ereal (integral\<^sup>L M (\<lambda>x. max 0 (f x)))"
by (subst integral_eq_nn_integral) (auto intro: real_of_ereal_pos nn_integral_nonneg assms)
also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = integral\<^sup>L M f"
using assms(2) by (intro integral_cong_AE assms integrable_max) auto
finally show ?thesis
by simp
qed
lemma integral_eq_zero_AE:
"f \<in> borel_measurable M \<Longrightarrow> (AE x in M. f x = 0) \<Longrightarrow> integral\<^sup>L M f = 0"
using integral_cong_AE[of f M "\<lambda>_. 0"] by simp
lemma integral_nonneg_eq_0_iff_AE:
fixes f :: "_ \<Rightarrow> real"
assumes f[measurable]: "integrable M f" and nonneg: "AE x in M. 0 \<le> f x"
shows "integral\<^sup>L M f = 0 \<longleftrightarrow> (AE x in M. f x = 0)"
proof
assume "integral\<^sup>L M f = 0"
then have "integral\<^sup>N M f = 0"
using nn_integral_eq_integral[OF f nonneg] by simp
then have "AE x in M. ereal (f x) \<le> 0"
by (simp add: nn_integral_0_iff_AE)
with nonneg show "AE x in M. f x = 0"
by auto
qed (auto simp add: integral_eq_zero_AE)
lemma integral_mono_AE:
fixes f :: "'a \<Rightarrow> real"
assumes "integrable M f" "integrable M g" "AE x in M. f x \<le> g x"
shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
proof -
have "0 \<le> integral\<^sup>L M (\<lambda>x. g x - f x)"
using assms by (intro integral_nonneg_AE integrable_diff assms) auto
also have "\<dots> = integral\<^sup>L M g - integral\<^sup>L M f"
by (intro integral_diff assms)
finally show ?thesis by simp
qed
lemma integral_mono:
fixes f :: "'a \<Rightarrow> real"
shows "integrable M f \<Longrightarrow> integrable M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x) \<Longrightarrow>
integral\<^sup>L M f \<le> integral\<^sup>L M g"
by (intro integral_mono_AE) auto
lemma (in finite_measure) integrable_measure:
assumes I: "disjoint_family_on X I" "countable I"
shows "integrable (count_space I) (\<lambda>i. measure M (X i))"
proof -
have "(\<integral>\<^sup>+i. measure M (X i) \<partial>count_space I) = (\<integral>\<^sup>+i. measure M (if X i \<in> sets M then X i else {}) \<partial>count_space I)"
by (auto intro!: nn_integral_cong measure_notin_sets)
also have "\<dots> = measure M (\<Union>i\<in>I. if X i \<in> sets M then X i else {})"
using I unfolding emeasure_eq_measure[symmetric]
by (subst emeasure_UN_countable) (auto simp: disjoint_family_on_def)
finally show ?thesis
by (auto intro!: integrableI_bounded simp: measure_nonneg)
qed
lemma integrableI_real_bounded:
assumes f: "f \<in> borel_measurable M" and ae: "AE x in M. 0 \<le> f x" and fin: "integral\<^sup>N M f < \<infinity>"
shows "integrable M f"
proof (rule integrableI_bounded)
have "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) = \<integral>\<^sup>+ x. ereal (f x) \<partial>M"
using ae by (auto intro: nn_integral_cong_AE)
also note fin
finally show "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>" .
qed fact
lemma integral_real_bounded:
assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "integral\<^sup>N M f \<le> ereal r"
shows "integral\<^sup>L M f \<le> r"
proof -
have "integrable M f"
using assms by (intro integrableI_real_bounded) auto
from nn_integral_eq_integral[OF this] assms show ?thesis
by simp
qed
subsection {* Restricted measure spaces *}
lemma integrable_restrict_space:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "integrable (restrict_space M \<Omega>) f \<longleftrightarrow> integrable M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
unfolding integrable_iff_bounded
borel_measurable_restrict_space_iff[OF \<Omega>]
nn_integral_restrict_space[OF \<Omega>]
by (simp add: ac_simps ereal_indicator times_ereal.simps(1)[symmetric] del: times_ereal.simps)
lemma integral_restrict_space:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes \<Omega>[simp]: "\<Omega> \<inter> space M \<in> sets M"
shows "integral\<^sup>L (restrict_space M \<Omega>) f = integral\<^sup>L M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
proof (rule integral_eq_cases)
assume "integrable (restrict_space M \<Omega>) f"
then show ?thesis
proof induct
case (base A c) then show ?case
by (simp add: indicator_inter_arith[symmetric] sets_restrict_space_iff
emeasure_restrict_space Int_absorb1 measure_restrict_space)
next
case (add g f) then show ?case
by (simp add: scaleR_add_right integrable_restrict_space)
next
case (lim f s)
show ?case
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> integral\<^sup>L (restrict_space M \<Omega>) f"
using lim by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) simp_all
show "(\<lambda>i. integral\<^sup>L (restrict_space M \<Omega>) (s i)) ----> (\<integral> x. indicator \<Omega> x *\<^sub>R f x \<partial>M)"
unfolding lim
using lim
by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (indicator \<Omega> x *\<^sub>R f x)"])
(auto simp add: space_restrict_space integrable_restrict_space
simp del: norm_scaleR
split: split_indicator)
qed
qed
qed (simp add: integrable_restrict_space)
subsection {* Measure spaces with an associated density *}
lemma integrable_density:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
assumes [measurable]: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
and nn: "AE x in M. 0 \<le> g x"
shows "integrable (density M g) f \<longleftrightarrow> integrable M (\<lambda>x. g x *\<^sub>R f x)"
unfolding integrable_iff_bounded using nn
apply (simp add: nn_integral_density )
apply (intro arg_cong2[where f="op ="] refl nn_integral_cong_AE)
apply auto
done
lemma integral_density:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
assumes f: "f \<in> borel_measurable M"
and g[measurable]: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
shows "integral\<^sup>L (density M g) f = integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
proof (rule integral_eq_cases)
assume "integrable (density M g) f"
then show ?thesis
proof induct
case (base A c)
then have [measurable]: "A \<in> sets M" by auto
have int: "integrable M (\<lambda>x. g x * indicator A x)"
using g base integrable_density[of "indicator A :: 'a \<Rightarrow> real" M g] by simp
then have "integral\<^sup>L M (\<lambda>x. g x * indicator A x) = (\<integral>\<^sup>+ x. ereal (g x * indicator A x) \<partial>M)"
using g by (subst nn_integral_eq_integral) auto
also have "\<dots> = (\<integral>\<^sup>+ x. ereal (g x) * indicator A x \<partial>M)"
by (intro nn_integral_cong) (auto split: split_indicator)
also have "\<dots> = emeasure (density M g) A"
by (rule emeasure_density[symmetric]) auto
also have "\<dots> = ereal (measure (density M g) A)"
using base by (auto intro: emeasure_eq_ereal_measure)
also have "\<dots> = integral\<^sup>L (density M g) (indicator A)"
using base by simp
finally show ?case
using base by (simp add: int)
next
case (add f h)
then have [measurable]: "f \<in> borel_measurable M" "h \<in> borel_measurable M"
by (auto dest!: borel_measurable_integrable)
from add g show ?case
by (simp add: scaleR_add_right integrable_density)
next
case (lim f s)
have [measurable]: "f \<in> borel_measurable M" "\<And>i. s i \<in> borel_measurable M"
using lim(1,5)[THEN borel_measurable_integrable] by auto
show ?case
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L M (\<lambda>x. g x *\<^sub>R f x)"
proof (rule integral_dominated_convergence)
show "integrable M (\<lambda>x. 2 * norm (g x *\<^sub>R f x))"
by (intro integrable_mult_right integrable_norm integrable_density[THEN iffD1] lim g) auto
show "AE x in M. (\<lambda>i. g x *\<^sub>R s i x) ----> g x *\<^sub>R f x"
using lim(3) by (auto intro!: tendsto_scaleR AE_I2[of M])
show "\<And>i. AE x in M. norm (g x *\<^sub>R s i x) \<le> 2 * norm (g x *\<^sub>R f x)"
using lim(4) g by (auto intro!: AE_I2[of M] mult_left_mono simp: field_simps)
qed auto
show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. g x *\<^sub>R s i x)) ----> integral\<^sup>L (density M g) f"
unfolding lim(2)[symmetric]
by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
(insert lim(3-5), auto)
qed
qed
qed (simp add: f g integrable_density)
lemma
fixes g :: "'a \<Rightarrow> real"
assumes "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x" "g \<in> borel_measurable M"
shows integral_real_density: "integral\<^sup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
and integrable_real_density: "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
using assms integral_density[of g M f] integrable_density[of g M f] by auto
lemma has_bochner_integral_density:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and g :: "'a \<Rightarrow> real"
shows "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (AE x in M. 0 \<le> g x) \<Longrightarrow>
has_bochner_integral M (\<lambda>x. g x *\<^sub>R f x) x \<Longrightarrow> has_bochner_integral (density M g) f x"
by (simp add: has_bochner_integral_iff integrable_density integral_density)
subsection {* Distributions *}
lemma integrable_distr_eq:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes [measurable]: "g \<in> measurable M N" "f \<in> borel_measurable N"
shows "integrable (distr M N g) f \<longleftrightarrow> integrable M (\<lambda>x. f (g x))"
unfolding integrable_iff_bounded by (simp_all add: nn_integral_distr)
lemma integrable_distr:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "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, where g=T])
(auto dest: borel_measurable_integrable)
lemma integral_distr:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes g[measurable]: "g \<in> measurable M N" and f: "f \<in> borel_measurable N"
shows "integral\<^sup>L (distr M N g) f = integral\<^sup>L M (\<lambda>x. f (g x))"
proof (rule integral_eq_cases)
assume "integrable (distr M N g) f"
then show ?thesis
proof induct
case (base A c)
then have [measurable]: "A \<in> sets N" by auto
from base have int: "integrable (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c)"
by (intro integrable_indicator)
have "integral\<^sup>L (distr M N g) (\<lambda>a. indicator A a *\<^sub>R c) = measure (distr M N g) A *\<^sub>R c"
using base by (subst integral_indicator) auto
also have "\<dots> = measure M (g -` A \<inter> space M) *\<^sub>R c"
by (subst measure_distr) auto
also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator (g -` A \<inter> space M) a *\<^sub>R c)"
using base by (subst integral_indicator) (auto simp: emeasure_distr)
also have "\<dots> = integral\<^sup>L M (\<lambda>a. indicator A (g a) *\<^sub>R c)"
using int base by (intro integral_cong_AE) (auto simp: emeasure_distr split: split_indicator)
finally show ?case .
next
case (add f h)
then have [measurable]: "f \<in> borel_measurable N" "h \<in> borel_measurable N"
by (auto dest!: borel_measurable_integrable)
from add g show ?case
by (simp add: scaleR_add_right integrable_distr_eq)
next
case (lim f s)
have [measurable]: "f \<in> borel_measurable N" "\<And>i. s i \<in> borel_measurable N"
using lim(1,5)[THEN borel_measurable_integrable] by auto
show ?case
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L M (\<lambda>x. f (g x))"
proof (rule integral_dominated_convergence)
show "integrable M (\<lambda>x. 2 * norm (f (g x)))"
using lim by (auto simp: integrable_distr_eq)
show "AE x in M. (\<lambda>i. s i (g x)) ----> f (g x)"
using lim(3) g[THEN measurable_space] by auto
show "\<And>i. AE x in M. norm (s i (g x)) \<le> 2 * norm (f (g x))"
using lim(4) g[THEN measurable_space] by auto
qed auto
show "(\<lambda>i. integral\<^sup>L M (\<lambda>x. s i (g x))) ----> integral\<^sup>L (distr M N g) f"
unfolding lim(2)[symmetric]
by (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
(insert lim(3-5), auto)
qed
qed
qed (simp add: f g integrable_distr_eq)
lemma has_bochner_integral_distr:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "f \<in> borel_measurable N \<Longrightarrow> g \<in> measurable M N \<Longrightarrow>
has_bochner_integral M (\<lambda>x. f (g x)) x \<Longrightarrow> has_bochner_integral (distr M N g) f x"
by (simp add: has_bochner_integral_iff integrable_distr_eq integral_distr)
subsection {* Lebesgue integration on @{const count_space} *}
lemma integrable_count_space:
fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
shows "finite X \<Longrightarrow> integrable (count_space X) f"
by (auto simp: nn_integral_count_space integrable_iff_bounded)
lemma measure_count_space[simp]:
"B \<subseteq> A \<Longrightarrow> finite B \<Longrightarrow> measure (count_space A) B = card B"
unfolding measure_def by (subst emeasure_count_space ) auto
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 eq: "\<And>x. x \<in> A \<Longrightarrow> (\<Sum>a | x = a \<and> a \<in> A \<and> f a \<noteq> 0. f a) = (\<Sum>x\<in>{x}. f x)"
by (intro setsum_mono_zero_cong_left) auto
have "(\<integral>x. f x \<partial>count_space A) = (\<integral>x. (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. indicator {a} x *\<^sub>R f a) \<partial>count_space A)"
by (intro integral_cong refl) (simp add: f eq)
also have "\<dots> = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. measure (count_space A) {a} *\<^sub>R f a)"
by (subst integral_setsum) (auto intro!: setsum_cong)
finally show ?thesis
by auto
qed
lemma lebesgue_integral_count_space_finite: "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
by (subst lebesgue_integral_count_space_finite_support)
(auto intro!: setsum_mono_zero_cong_left)
subsection {* Point measure *}
lemma lebesgue_integral_point_measure_finite:
fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "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 *\<^sub>R g a)"
by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
lemma integrable_point_measure_finite:
fixes g :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}" and f :: "'a \<Rightarrow> real"
shows "finite A \<Longrightarrow> integrable (point_measure A f) g"
unfolding point_measure_def
apply (subst density_ereal_max_0)
apply (subst integrable_density)
apply (auto simp: AE_count_space integrable_count_space)
done
subsection {* Legacy lemmas for the real-valued Lebesgue integral *}
lemma real_lebesgue_integral_def:
assumes f: "integrable M f"
shows "integral\<^sup>L M f = real (\<integral>\<^sup>+x. f x \<partial>M) - real (\<integral>\<^sup>+x. - f x \<partial>M)"
proof -
have "integral\<^sup>L M f = integral\<^sup>L M (\<lambda>x. max 0 (f x) - max 0 (- f x))"
by (auto intro!: arg_cong[where f="integral\<^sup>L M"])
also have "\<dots> = integral\<^sup>L M (\<lambda>x. max 0 (f x)) - integral\<^sup>L M (\<lambda>x. max 0 (- f x))"
by (intro integral_diff integrable_max integrable_minus integrable_zero f)
also have "integral\<^sup>L M (\<lambda>x. max 0 (f x)) = real (\<integral>\<^sup>+x. max 0 (f x) \<partial>M)"
by (subst integral_eq_nn_integral[symmetric]) (auto intro!: f)
also have "integral\<^sup>L M (\<lambda>x. max 0 (- f x)) = real (\<integral>\<^sup>+x. max 0 (- f x) \<partial>M)"
by (subst integral_eq_nn_integral[symmetric]) (auto intro!: f)
also have "(\<lambda>x. ereal (max 0 (f x))) = (\<lambda>x. max 0 (ereal (f x)))"
by (auto simp: max_def)
also have "(\<lambda>x. ereal (max 0 (- f x))) = (\<lambda>x. max 0 (- ereal (f x)))"
by (auto simp: max_def)
finally show ?thesis
unfolding nn_integral_max_0 .
qed
lemma real_integrable_def:
"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>"
unfolding integrable_iff_bounded
proof (safe del: notI)
assume *: "(\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M) < \<infinity>"
have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
by (intro nn_integral_mono) auto
also note *
finally show "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>"
by simp
have "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<le> (\<integral>\<^sup>+ x. ereal (norm (f x)) \<partial>M)"
by (intro nn_integral_mono) auto
also note *
finally show "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
by simp
next
assume [measurable]: "f \<in> borel_measurable M"
assume fin: "(\<integral>\<^sup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^sup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
have "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) + max 0 (ereal (- f x)) \<partial>M)"
by (intro nn_integral_cong) (auto simp: max_def)
also have"\<dots> = (\<integral>\<^sup>+ x. max 0 (ereal (f x)) \<partial>M) + (\<integral>\<^sup>+ x. max 0 (ereal (- f x)) \<partial>M)"
by (intro nn_integral_add) auto
also have "\<dots> < \<infinity>"
using fin by (auto simp: nn_integral_max_0)
finally show "(\<integral>\<^sup>+ x. norm (f x) \<partial>M) < \<infinity>" .
qed
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 real_integrable_def by auto
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 real_integrable_def real_lebesgue_integral_def[OF assms]
using nn_integral_nonneg[of M "\<lambda>x. ereal (f x)"]
using nn_integral_nonneg[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_monotone_convergence_nonneg:
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
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 nn_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 nn_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 nn_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 nn_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: real_integrable_def real_lebesgue_integral_def u)
qed
lemma
fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"
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_monotone_convergence: "integrable M u"
and integral_monotone_convergence: "integral\<^sup>L M u = x"
and has_bochner_integral_monotone_convergence: "has_bochner_integral 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_nonneg[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 integrable_add)
with diff(2) f show "integrable M u" "integral\<^sup>L M u = x"
by auto
then show "has_bochner_integral M u x"
by (metis has_bochner_integral_integrable)
qed
lemma integral_norm_eq_0_iff:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes f[measurable]: "integrable M f"
shows "(\<integral>x. norm (f x) \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
proof -
have "(\<integral>\<^sup>+x. norm (f x) \<partial>M) = (\<integral>x. norm (f x) \<partial>M)"
using f by (intro nn_integral_eq_integral integrable_norm) auto
then have "(\<integral>x. norm (f x) \<partial>M) = 0 \<longleftrightarrow> (\<integral>\<^sup>+x. norm (f x) \<partial>M) = 0"
by simp
also have "\<dots> \<longleftrightarrow> emeasure M {x\<in>space M. ereal (norm (f x)) \<noteq> 0} = 0"
by (intro nn_integral_0_iff) auto
finally show ?thesis
by simp
qed
lemma integral_0_iff:
fixes f :: "'a \<Rightarrow> real"
shows "integrable M f \<Longrightarrow> (\<integral>x. abs (f x) \<partial>M) = 0 \<longleftrightarrow> emeasure M {x\<in>space M. f x \<noteq> 0} = 0"
using integral_norm_eq_0_iff[of M f] by simp
lemma (in finite_measure) lebesgue_integral_const[simp]:
"(\<integral>x. a \<partial>M) = measure M (space M) *\<^sub>R a"
using integral_indicator[of "space M" M a]
by (simp del: integral_indicator integral_scaleR_left cong: integral_cong)
lemma (in finite_measure) integrable_const[intro!, simp]: "integrable M (\<lambda>x. a)"
using integrable_indicator[of "space M" M a] by (simp cong: integrable_cong)
lemma (in finite_measure) integrable_const_bound:
fixes f :: "'a \<Rightarrow> 'b::{banach,second_countable_topology}"
shows "AE x in M. norm (f x) \<le> B \<Longrightarrow> f \<in> borel_measurable M \<Longrightarrow> integrable M f"
apply (rule integrable_bound[OF integrable_const[of B], of f])
apply assumption
apply (cases "0 \<le> B")
apply auto
done
lemma (in finite_measure) integral_less_AE:
fixes X Y :: "'a \<Rightarrow> real"
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 int 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 nn_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:
fixes X Y :: "'a \<Rightarrow> real"
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 integrable_mult_indicator:
fixes f :: "'a \<Rightarrow> real"
shows "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 split: split_indicator)
lemma tendsto_integral_at_top:
fixes f :: "real \<Rightarrow> real"
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 :: "real \<Rightarrow> real" assume f: "integrable M f" "\<And>x. 0 \<le> f x"
then have [measurable]: "f \<in> borel_measurable borel"
by (simp add: real_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 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)
show "\<And>j. AE x in M. norm (f x * indicator {.. j} x) \<le> f x"
using f(2) by (intro AE_I2) (auto split: split_indicator)
qed (simp | fact)+
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 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[symmetric])
(auto intro!: integrable_mult_indicator f integral_cong
simp: M split: split_max)
also have "?p - ?n = integral\<^sup>L M f"
by (subst integral_diff[symmetric]) (auto intro!: f integral_cong simp: M split: split_max)
finally show ?thesis .
qed
lemma
fixes f :: "real \<Rightarrow> real"
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 has_bochner_integral_monotone_convergence_at_top: "has_bochner_integral M f x"
and integrable_monotone_convergence_at_top: "integrable M f"
and integral_monotone_convergence_at_top:"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 "has_bochner_integral M f x"
by (rule has_bochner_integral_monotone_convergence) fact+
then show "integrable M f" "integral\<^sup>L M f = x"
by (auto simp: _has_bochner_integral_iff)
qed
subsection {* Product measure *}
lemma (in sigma_finite_measure) borel_measurable_lebesgue_integrable[measurable (raw)]:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
assumes [measurable]: "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
shows "Measurable.pred N (\<lambda>x. integrable M (f x))"
proof -
have [simp]: "\<And>x. x \<in> space N \<Longrightarrow> integrable M (f x) \<longleftrightarrow> (\<integral>\<^sup>+y. norm (f x y) \<partial>M) < \<infinity>"
unfolding integrable_iff_bounded by simp
show ?thesis
by (simp cong: measurable_cong)
qed
lemma (in sigma_finite_measure) measurable_measure[measurable (raw)]:
"(\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M) \<Longrightarrow>
{x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M) \<Longrightarrow>
(\<lambda>x. measure M (A x)) \<in> borel_measurable N"
unfolding measure_def by (intro measurable_emeasure borel_measurable_real_of_ereal)
lemma Collect_subset [simp]: "{x\<in>A. P x} \<subseteq> A" by auto
lemma (in sigma_finite_measure) borel_measurable_lebesgue_integral[measurable (raw)]:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f[measurable]: "split f \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
shows "(\<lambda>x. \<integral>y. f x y \<partial>M) \<in> borel_measurable N"
proof -
from borel_measurable_implies_sequence_metric[OF f, of 0] guess s ..
then have s: "\<And>i. simple_function (N \<Otimes>\<^sub>M M) (s i)"
"\<And>x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> (\<lambda>i. s i (x, y)) ----> f x y"
"\<And>i x y. x \<in> space N \<Longrightarrow> y \<in> space M \<Longrightarrow> norm (s i (x, y)) \<le> 2 * norm (f x y)"
by (auto simp: space_pair_measure norm_conv_dist)
have [measurable]: "\<And>i. s i \<in> borel_measurable (N \<Otimes>\<^sub>M M)"
by (rule borel_measurable_simple_function) fact
have "\<And>i. s i \<in> measurable (N \<Otimes>\<^sub>M M) (count_space UNIV)"
by (rule measurable_simple_function) fact
def f' \<equiv> "\<lambda>i x. if integrable M (f x)
then simple_bochner_integral M (\<lambda>y. s i (x, y))
else (THE x. False)"
{ fix i x assume "x \<in> space N"
then have "simple_bochner_integral M (\<lambda>y. s i (x, y)) =
(\<Sum>z\<in>s i ` (space N \<times> space M). measure M {y \<in> space M. s i (x, y) = z} *\<^sub>R z)"
using s(1)[THEN simple_functionD(1)]
unfolding simple_bochner_integral_def
by (intro setsum_mono_zero_cong_left)
(auto simp: eq_commute space_pair_measure image_iff cong: conj_cong) }
note eq = this
show ?thesis
proof (rule borel_measurable_LIMSEQ_metric)
fix i show "f' i \<in> borel_measurable N"
unfolding f'_def by (simp_all add: eq cong: measurable_cong if_cong)
next
fix x assume x: "x \<in> space N"
{ assume int_f: "integrable M (f x)"
have int_2f: "integrable M (\<lambda>y. 2 * norm (f x y))"
by (intro integrable_norm integrable_mult_right int_f)
have "(\<lambda>i. integral\<^sup>L M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
proof (rule integral_dominated_convergence)
from int_f show "f x \<in> borel_measurable M" by auto
show "\<And>i. (\<lambda>y. s i (x, y)) \<in> borel_measurable M"
using x by simp
show "AE xa in M. (\<lambda>i. s i (x, xa)) ----> f x xa"
using x s(2) by auto
show "\<And>i. AE xa in M. norm (s i (x, xa)) \<le> 2 * norm (f x xa)"
using x s(3) by auto
qed fact
moreover
{ fix i
have "simple_bochner_integrable M (\<lambda>y. s i (x, y))"
proof (rule simple_bochner_integrableI_bounded)
have "(\<lambda>y. s i (x, y)) ` space M \<subseteq> s i ` (space N \<times> space M)"
using x by auto
then show "simple_function M (\<lambda>y. s i (x, y))"
using simple_functionD(1)[OF s(1), of i] x
by (intro simple_function_borel_measurable)
(auto simp: space_pair_measure dest: finite_subset)
have "(\<integral>\<^sup>+ y. ereal (norm (s i (x, y))) \<partial>M) \<le> (\<integral>\<^sup>+ y. 2 * norm (f x y) \<partial>M)"
using x s by (intro nn_integral_mono) auto
also have "(\<integral>\<^sup>+ y. 2 * norm (f x y) \<partial>M) < \<infinity>"
using int_2f by (simp add: integrable_iff_bounded)
finally show "(\<integral>\<^sup>+ xa. ereal (norm (s i (x, xa))) \<partial>M) < \<infinity>" .
qed
then have "integral\<^sup>L M (\<lambda>y. s i (x, y)) = simple_bochner_integral M (\<lambda>y. s i (x, y))"
by (rule simple_bochner_integrable_eq_integral[symmetric]) }
ultimately have "(\<lambda>i. simple_bochner_integral M (\<lambda>y. s i (x, y))) ----> integral\<^sup>L M (f x)"
by simp }
then
show "(\<lambda>i. f' i x) ----> integral\<^sup>L M (f x)"
unfolding f'_def
by (cases "integrable M (f x)") (simp_all add: not_integrable_integral_eq tendsto_const)
qed
qed
lemma (in pair_sigma_finite) integrable_product_swap:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes "integrable (M1 \<Otimes>\<^sub>M M2) f"
shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x))"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
show ?thesis unfolding *
by (rule integrable_distr[OF measurable_pair_swap'])
(simp add: distr_pair_swap[symmetric] assms)
qed
lemma (in pair_sigma_finite) integrable_product_swap_iff:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x,y). f (y,x)) \<longleftrightarrow> integrable (M1 \<Otimes>\<^sub>M M2) f"
proof -
interpret Q: pair_sigma_finite M2 M1 by default
from Q.integrable_product_swap[of "\<lambda>(x,y). f (y,x)"] integrable_product_swap[of f]
show ?thesis by auto
qed
lemma (in pair_sigma_finite) integral_product_swap:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
shows "(\<integral>(x,y). f (y,x) \<partial>(M2 \<Otimes>\<^sub>M M1)) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
proof -
have *: "(\<lambda>(x,y). f (y,x)) = (\<lambda>x. f (case x of (x,y)\<Rightarrow>(y,x)))" by (auto simp: fun_eq_iff)
show ?thesis unfolding *
by (simp add: integral_distr[symmetric, OF measurable_pair_swap' f] distr_pair_swap[symmetric])
qed
lemma (in pair_sigma_finite) emeasure_pair_measure_finite:
assumes A: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" and finite: "emeasure (M1 \<Otimes>\<^sub>M M2) A < \<infinity>"
shows "AE x in M1. emeasure M2 {y\<in>space M2. (x, y) \<in> A} < \<infinity>"
proof -
from M2.emeasure_pair_measure_alt[OF A] finite
have "(\<integral>\<^sup>+ x. emeasure M2 (Pair x -` A) \<partial>M1) \<noteq> \<infinity>"
by simp
then have "AE x in M1. emeasure M2 (Pair x -` A) \<noteq> \<infinity>"
by (rule nn_integral_PInf_AE[rotated]) (intro M2.measurable_emeasure_Pair A)
moreover have "\<And>x. x \<in> space M1 \<Longrightarrow> Pair x -` A = {y\<in>space M2. (x, y) \<in> A}"
using sets.sets_into_space[OF A] by (auto simp: space_pair_measure)
ultimately show ?thesis by auto
qed
lemma (in pair_sigma_finite) AE_integrable_fst':
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
shows "AE x in M1. integrable M2 (\<lambda>y. f (x, y))"
proof -
have "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1) = (\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2))"
by (rule M2.nn_integral_fst) simp
also have "(\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2)) \<noteq> \<infinity>"
using f unfolding integrable_iff_bounded by simp
finally have "AE x in M1. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<noteq> \<infinity>"
by (intro nn_integral_PInf_AE M2.borel_measurable_nn_integral )
(auto simp: measurable_split_conv)
with AE_space show ?thesis
by eventually_elim
(auto simp: integrable_iff_bounded measurable_compose[OF _ borel_measurable_integrable[OF f]])
qed
lemma (in pair_sigma_finite) integrable_fst':
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) f"
shows "integrable M1 (\<lambda>x. \<integral>y. f (x, y) \<partial>M2)"
unfolding integrable_iff_bounded
proof
show "(\<lambda>x. \<integral> y. f (x, y) \<partial>M2) \<in> borel_measurable M1"
by (rule M2.borel_measurable_lebesgue_integral) simp
have "(\<integral>\<^sup>+ x. ereal (norm (\<integral> y. f (x, y) \<partial>M2)) \<partial>M1) \<le> (\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1)"
using AE_integrable_fst'[OF f] by (auto intro!: nn_integral_mono_AE integral_norm_bound_ereal)
also have "(\<integral>\<^sup>+x. (\<integral>\<^sup>+y. norm (f (x, y)) \<partial>M2) \<partial>M1) = (\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2))"
by (rule M2.nn_integral_fst) simp
also have "(\<integral>\<^sup>+x. norm (f x) \<partial>(M1 \<Otimes>\<^sub>M M2)) < \<infinity>"
using f unfolding integrable_iff_bounded by simp
finally show "(\<integral>\<^sup>+ x. ereal (norm (\<integral> y. f (x, y) \<partial>M2)) \<partial>M1) < \<infinity>" .
qed
lemma (in pair_sigma_finite) integral_fst':
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) f"
shows "(\<integral>x. (\<integral>y. f (x, y) \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
using f proof induct
case (base A c)
have A[measurable]: "A \<in> sets (M1 \<Otimes>\<^sub>M M2)" by fact
have eq: "\<And>x y. x \<in> space M1 \<Longrightarrow> indicator A (x, y) = indicator {y\<in>space M2. (x, y) \<in> A} y"
using sets.sets_into_space[OF A] by (auto split: split_indicator simp: space_pair_measure)
have int_A: "integrable (M1 \<Otimes>\<^sub>M M2) (indicator A :: _ \<Rightarrow> real)"
using base by (rule integrable_real_indicator)
have "(\<integral> x. \<integral> y. indicator A (x, y) *\<^sub>R c \<partial>M2 \<partial>M1) = (\<integral>x. measure M2 {y\<in>space M2. (x, y) \<in> A} *\<^sub>R c \<partial>M1)"
proof (intro integral_cong_AE, simp, simp)
from AE_integrable_fst'[OF int_A] AE_space
show "AE x in M1. (\<integral>y. indicator A (x, y) *\<^sub>R c \<partial>M2) = measure M2 {y\<in>space M2. (x, y) \<in> A} *\<^sub>R c"
by eventually_elim (simp add: eq integrable_indicator_iff)
qed
also have "\<dots> = measure (M1 \<Otimes>\<^sub>M M2) A *\<^sub>R c"
proof (subst integral_scaleR_left)
have "(\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1) =
(\<integral>\<^sup>+x. emeasure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1)"
using emeasure_pair_measure_finite[OF base]
by (intro nn_integral_cong_AE, eventually_elim) (simp add: emeasure_eq_ereal_measure)
also have "\<dots> = emeasure (M1 \<Otimes>\<^sub>M M2) A"
using sets.sets_into_space[OF A]
by (subst M2.emeasure_pair_measure_alt)
(auto intro!: nn_integral_cong arg_cong[where f="emeasure M2"] simp: space_pair_measure)
finally have *: "(\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1) = emeasure (M1 \<Otimes>\<^sub>M M2) A" .
from base * show "integrable M1 (\<lambda>x. measure M2 {y \<in> space M2. (x, y) \<in> A})"
by (simp add: measure_nonneg integrable_iff_bounded)
then have "(\<integral>x. measure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1) =
(\<integral>\<^sup>+x. ereal (measure M2 {y \<in> space M2. (x, y) \<in> A}) \<partial>M1)"
by (rule nn_integral_eq_integral[symmetric]) (simp add: measure_nonneg)
also note *
finally show "(\<integral>x. measure M2 {y \<in> space M2. (x, y) \<in> A} \<partial>M1) *\<^sub>R c = measure (M1 \<Otimes>\<^sub>M M2) A *\<^sub>R c"
using base by (simp add: emeasure_eq_ereal_measure)
qed
also have "\<dots> = (\<integral> a. indicator A a *\<^sub>R c \<partial>(M1 \<Otimes>\<^sub>M M2))"
using base by simp
finally show ?case .
next
case (add f g)
then have [measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" "g \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
by auto
have "(\<integral> x. \<integral> y. f (x, y) + g (x, y) \<partial>M2 \<partial>M1) =
(\<integral> x. (\<integral> y. f (x, y) \<partial>M2) + (\<integral> y. g (x, y) \<partial>M2) \<partial>M1)"
apply (rule integral_cong_AE)
apply simp_all
using AE_integrable_fst'[OF add(1)] AE_integrable_fst'[OF add(3)]
apply eventually_elim
apply simp
done
also have "\<dots> = (\<integral> x. f x \<partial>(M1 \<Otimes>\<^sub>M M2)) + (\<integral> x. g x \<partial>(M1 \<Otimes>\<^sub>M M2))"
using integrable_fst'[OF add(1)] integrable_fst'[OF add(3)] add(2,4) by simp
finally show ?case
using add by simp
next
case (lim f s)
then have [measurable]: "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" "\<And>i. s i \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
by auto
show ?case
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
proof (rule integral_dominated_convergence)
show "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. 2 * norm (f x))"
using lim(5) by auto
qed (insert lim, auto)
have "(\<lambda>i. \<integral> x. \<integral> y. s i (x, y) \<partial>M2 \<partial>M1) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
proof (rule integral_dominated_convergence)
have "AE x in M1. \<forall>i. integrable M2 (\<lambda>y. s i (x, y))"
unfolding AE_all_countable using AE_integrable_fst'[OF lim(1)] ..
with AE_space AE_integrable_fst'[OF lim(5)]
show "AE x in M1. (\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
proof eventually_elim
fix x assume x: "x \<in> space M1" and
s: "\<forall>i. integrable M2 (\<lambda>y. s i (x, y))" and f: "integrable M2 (\<lambda>y. f (x, y))"
show "(\<lambda>i. \<integral> y. s i (x, y) \<partial>M2) ----> \<integral> y. f (x, y) \<partial>M2"
proof (rule integral_dominated_convergence)
show "integrable M2 (\<lambda>y. 2 * norm (f (x, y)))"
using f by auto
show "AE xa in M2. (\<lambda>i. s i (x, xa)) ----> f (x, xa)"
using x lim(3) by (auto simp: space_pair_measure)
show "\<And>i. AE xa in M2. norm (s i (x, xa)) \<le> 2 * norm (f (x, xa))"
using x lim(4) by (auto simp: space_pair_measure)
qed (insert x, measurable)
qed
show "integrable M1 (\<lambda>x. (\<integral> y. 2 * norm (f (x, y)) \<partial>M2))"
by (intro integrable_mult_right integrable_norm integrable_fst' lim)
fix i show "AE x in M1. norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral> y. 2 * norm (f (x, y)) \<partial>M2)"
using AE_space AE_integrable_fst'[OF lim(1), of i] AE_integrable_fst'[OF lim(5)]
proof eventually_elim
fix x assume x: "x \<in> space M1"
and s: "integrable M2 (\<lambda>y. s i (x, y))" and f: "integrable M2 (\<lambda>y. f (x, y))"
from s have "norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral>\<^sup>+y. norm (s i (x, y)) \<partial>M2)"
by (rule integral_norm_bound_ereal)
also have "\<dots> \<le> (\<integral>\<^sup>+y. 2 * norm (f (x, y)) \<partial>M2)"
using x lim by (auto intro!: nn_integral_mono simp: space_pair_measure)
also have "\<dots> = (\<integral>y. 2 * norm (f (x, y)) \<partial>M2)"
using f by (intro nn_integral_eq_integral) auto
finally show "norm (\<integral> y. s i (x, y) \<partial>M2) \<le> (\<integral> y. 2 * norm (f (x, y)) \<partial>M2)"
by simp
qed
qed simp_all
then show "(\<lambda>i. integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (s i)) ----> \<integral> x. \<integral> y. f (x, y) \<partial>M2 \<partial>M1"
using lim by simp
qed
qed
lemma (in pair_sigma_finite)
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
shows AE_integrable_fst: "AE x in M1. integrable M2 (\<lambda>y. f x y)" (is "?AE")
and integrable_fst: "integrable M1 (\<lambda>x. \<integral>y. f x y \<partial>M2)" (is "?INT")
and integral_fst: "(\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" (is "?EQ")
using AE_integrable_fst'[OF f] integrable_fst'[OF f] integral_fst'[OF f] by auto
lemma (in pair_sigma_finite)
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _::{banach, second_countable_topology}"
assumes f[measurable]: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
shows AE_integrable_snd: "AE y in M2. integrable M1 (\<lambda>x. f x y)" (is "?AE")
and integrable_snd: "integrable M2 (\<lambda>y. \<integral>x. f x y \<partial>M1)" (is "?INT")
and integral_snd: "(\<integral>y. (\<integral>x. f x y \<partial>M1) \<partial>M2) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) (split f)" (is "?EQ")
proof -
interpret Q: pair_sigma_finite M2 M1 by default
have Q_int: "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f y x)"
using f unfolding integrable_product_swap_iff[symmetric] by simp
show ?AE using Q.AE_integrable_fst'[OF Q_int] by simp
show ?INT using Q.integrable_fst'[OF Q_int] by simp
show ?EQ using Q.integral_fst'[OF Q_int]
using integral_product_swap[of "split f"] by simp
qed
lemma (in pair_sigma_finite) Fubini_integral:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes f: "integrable (M1 \<Otimes>\<^sub>M M2) (split f)"
shows "(\<integral>y. (\<integral>x. f x y \<partial>M1) \<partial>M2) = (\<integral>x. (\<integral>y. f x y \<partial>M2) \<partial>M1)"
unfolding integral_snd[OF assms] integral_fst[OF assms] ..
lemma (in product_sigma_finite) product_integral_singleton:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
shows "f \<in> borel_measurable (M i) \<Longrightarrow> (\<integral>x. f (x i) \<partial>Pi\<^sub>M {i} M) = integral\<^sup>L (M i) f"
apply (subst distr_singleton[symmetric])
apply (subst integral_distr)
apply simp_all
done
lemma (in product_sigma_finite) product_integral_fold:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
and f: "integrable (Pi\<^sub>M (I \<union> J) M) f"
shows "integral\<^sup>L (Pi\<^sub>M (I \<union> J) M) f = (\<integral>x. (\<integral>y. f (merge I J (x, y)) \<partial>Pi\<^sub>M J M) \<partial>Pi\<^sub>M I M)"
proof -
interpret I: finite_product_sigma_finite M I by default fact
interpret J: finite_product_sigma_finite M J by default fact
have "finite (I \<union> J)" using fin by auto
interpret IJ: finite_product_sigma_finite M "I \<union> J" by default fact
interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by default
let ?M = "merge I J"
let ?f = "\<lambda>x. f (?M x)"
from f have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
by auto
have P_borel: "(\<lambda>x. f (merge I J x)) \<in> borel_measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)"
using measurable_comp[OF measurable_merge f_borel] by (simp add: comp_def)
have f_int: "integrable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) ?f"
by (rule integrable_distr[OF measurable_merge]) (simp add: distr_merge[OF IJ fin] f)
show ?thesis
apply (subst distr_merge[symmetric, OF IJ fin])
apply (subst integral_distr[OF measurable_merge f_borel])
apply (subst P.integral_fst'[symmetric, OF f_int])
apply simp
done
qed
lemma (in product_sigma_finite) product_integral_insert:
fixes f :: "_ \<Rightarrow> _::{banach, second_countable_topology}"
assumes I: "finite I" "i \<notin> I"
and f: "integrable (Pi\<^sub>M (insert i I) M) f"
shows "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = (\<integral>x. (\<integral>y. f (x(i:=y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
proof -
have "integral\<^sup>L (Pi\<^sub>M (insert i I) M) f = integral\<^sup>L (Pi\<^sub>M (I \<union> {i}) M) f"
by simp
also have "\<dots> = (\<integral>x. (\<integral>y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) \<partial>Pi\<^sub>M I M)"
using f I by (intro product_integral_fold) auto
also have "\<dots> = (\<integral>x. (\<integral>y. f (x(i := y)) \<partial>M i) \<partial>Pi\<^sub>M I M)"
proof (rule integral_cong[OF refl], subst product_integral_singleton[symmetric])
fix x assume x: "x \<in> space (Pi\<^sub>M I M)"
have f_borel: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
using f by auto
show "(\<lambda>y. f (x(i := y))) \<in> borel_measurable (M i)"
using measurable_comp[OF measurable_component_update f_borel, OF x `i \<notin> I`]
unfolding comp_def .
from x I show "(\<integral> y. f (merge I {i} (x,y)) \<partial>Pi\<^sub>M {i} M) = (\<integral> xa. f (x(i := xa i)) \<partial>Pi\<^sub>M {i} M)"
by (auto intro!: integral_cong arg_cong[where f=f] simp: merge_def space_PiM extensional_def PiE_def)
qed
finally show ?thesis .
qed
lemma (in product_sigma_finite) product_integrable_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
assumes [simp]: "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "integrable (Pi\<^sub>M I M) (\<lambda>x. (\<Prod>i\<in>I. f i (x i)))" (is "integrable _ ?f")
proof (unfold integrable_iff_bounded, intro conjI)
interpret finite_product_sigma_finite M I by default fact
show "?f \<in> borel_measurable (Pi\<^sub>M I M)"
using assms by simp
have "(\<integral>\<^sup>+ x. ereal (norm (\<Prod>i\<in>I. f i (x i))) \<partial>Pi\<^sub>M I M) =
(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. ereal (norm (f i (x i)))) \<partial>Pi\<^sub>M I M)"
by (simp add: setprod_norm setprod_ereal)
also have "\<dots> = (\<Prod>i\<in>I. \<integral>\<^sup>+ x. ereal (norm (f i x)) \<partial>M i)"
using assms by (intro product_nn_integral_setprod) auto
also have "\<dots> < \<infinity>"
using integrable by (simp add: setprod_PInf nn_integral_nonneg integrable_iff_bounded)
finally show "(\<integral>\<^sup>+ x. ereal (norm (\<Prod>i\<in>I. f i (x i))) \<partial>Pi\<^sub>M I M) < \<infinity>" .
qed
lemma (in product_sigma_finite) product_integral_setprod:
fixes f :: "'i \<Rightarrow> 'a \<Rightarrow> _::{real_normed_field,banach,second_countable_topology}"
assumes "finite I" and integrable: "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)"
shows "(\<integral>x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>L (M i) (f i))"
using assms proof induct
case empty
interpret finite_measure "Pi\<^sub>M {} M"
by rule (simp add: space_PiM)
show ?case by (simp add: space_PiM measure_def)
next
case (insert i I)
then have iI: "finite (insert i I)" by auto
then have prod: "\<And>J. J \<subseteq> insert i I \<Longrightarrow>
integrable (Pi\<^sub>M J M) (\<lambda>x. (\<Prod>i\<in>J. f i (x i)))"
by (intro product_integrable_setprod insert(4)) (auto intro: finite_subset)
interpret I: finite_product_sigma_finite M I by default fact
have *: "\<And>x y. (\<Prod>j\<in>I. f j (if j = i then y else x j)) = (\<Prod>j\<in>I. f j (x j))"
using `i \<notin> I` by (auto intro!: setprod_cong)
show ?case
unfolding product_integral_insert[OF insert(1,2) prod[OF subset_refl]]
by (simp add: * insert prod subset_insertI)
qed
lemma integrable_subalgebra:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
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)
proof -
have "f \<in> borel_measurable M"
using assms by (auto simp: measurable_def)
with assms show ?thesis
using assms by (auto simp: integrable_iff_bounded nn_integral_subalgebra)
qed
lemma integral_subalgebra:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
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 "integral\<^sup>L N f = integral\<^sup>L M f"
proof cases
assume "integrable N f"
then show ?thesis
proof induct
case base with assms show ?case by (auto simp: subset_eq measure_def)
next
case (add f g)
then have "(\<integral> a. f a + g a \<partial>N) = integral\<^sup>L M f + integral\<^sup>L M g"
by simp
also have "\<dots> = (\<integral> a. f a + g a \<partial>M)"
using add integrable_subalgebra[OF _ N, of f] integrable_subalgebra[OF _ N, of g] by simp
finally show ?case .
next
case (lim f s)
then have M: "\<And>i. integrable M (s i)" "integrable M f"
using integrable_subalgebra[OF _ N, of f] integrable_subalgebra[OF _ N, of "s i" for i] by simp_all
show ?case
proof (intro LIMSEQ_unique)
show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L N f"
apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
using lim
apply auto
done
show "(\<lambda>i. integral\<^sup>L N (s i)) ----> integral\<^sup>L M f"
unfolding lim
apply (rule integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"])
using lim M N(2)
apply auto
done
qed
qed
qed (simp add: not_integrable_integral_eq integrable_subalgebra[OF assms])
hide_const simple_bochner_integral
hide_const simple_bochner_integrable
end