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Quellcode-Bibliothek
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Datei:
denumerable_enumeration.prf
Sprache: Isabelle
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(* Title: HOL/Analysis/Bochner_Integration.thy
Author: Johannes Hölzl, TU München
*)
section \<open>Bochner Integration for Vector-Valued Functions\<close>
theory Bochner_Integration
imports Finite_Product_Measure
begin
text \<open>
In the following development of the Bochner integral we use second countable topologies instead
of separable spaces. A second countable topology is also separable.
\<close>
proposition borel_measurable_implies_sequence_metric:
fixes f :: "'a \ 'b :: {metric_space, second_countable_topology}"
assumes [measurable]: "f \ borel_measurable M"
shows "\F. (\i. simple_function M (F i)) \ (\x\space M. (\i. F i x) \ f x) \
(\<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: "\X. open X \ X \ {} \ \d\D. d \ X"
by (erule countable_dense_setE)
define e where "e = from_nat_into D"
{ fix n x
obtain d where "d \ D" and d: "d \ ball x (1 / Suc n)"
using D[of "ball x (1 / Suc n)"] by auto
from \<open>d \<in> D\<close> D[of UNIV] \<open>countable D\<close> obtain i where "d = e i"
unfolding e_def by (auto dest: from_nat_into_surj)
with d have "\i. dist x (e i) < 1 / Suc n"
by auto }
note e = this
define A where [abs_def]: "A m n =
{x\<in>space M. dist (f x) (e n) < 1 / (Suc m) \<and> 1 / (Suc m) \<le> dist (f x) z}" for m n
define B where [abs_def]: "B m = disjointed (A m)" for m
define m where [abs_def]: "m N x = Max {m. m \ N \ x \ (\n\N. B m n)}" for N x
define F where [abs_def]: "F N 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)" for N x
have B_imp_A[intro, simp]: "\x m n. x \ B m n \ x \ A m n"
using disjointed_subset[of "A m" for m] unfolding B_def by auto
{ fix m
have "\n. A m n \ sets M"
by (auto simp: A_def)
then have "\n. B m n \ sets M"
using sets.range_disjointed_sets[of "A m" M] by (auto simp: B_def) }
note this[measurable]
{ fix N i x assume "\m\N. x \ (\n\N. B m n)"
then have "m N x \ {m::nat. m \ N \ x \ (\n\N. B m n)}"
unfolding m_def by (intro Max_in) auto
then have "m N x \ N" "\n\N. x \ B (m N x) n"
by auto }
note m = this
{ fix j N i x assume "j \ N" "i \ N" "x \ B j i"
then have "j \ 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]: "\i. F i \ borel_measurable M"
unfolding F_def m_def by measurable
show "\x i. F i -` {x} \ space M \ sets M"
by measurable
{ fix i
{ fix n x assume "x \ B (m i x) n"
then have "(LEAST n. x \ B (m i x) n) \ n"
by (intro Least_le)
also assume "n \ i"
finally have "(LEAST n. x \ B (m i x) n) \ i" . }
then have "F i ` space M \ {z} \ 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 \ N" "n \ N" "x \ B i n"
then have 1: "\m\N. x \ (\n\N. B m n)" by auto
from m[OF this] obtain n where n: "m N x \ N" "n \ N" "x \ B (m N x) n" by auto
moreover
define L where "L = (LEAST n. x \ B (m N x) n)"
have "dist (f x) (e L) < 1 / Suc (m N x)"
proof -
have "x \ B (m N x) L"
using n(3) unfolding L_def by (rule LeastI)
then have "x \ 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 \ space M"
show "(\i. F i x) \ f x"
proof cases
assume "f x = z"
then have "\i n. x \ A i n"
unfolding A_def by auto
then have "\i. F i x = z"
by (auto simp: F_def)
then show ?thesis
using \<open>f x = z\<close> by auto
next
assume "f x \ z"
show ?thesis
proof (rule tendstoI)
fix e :: real assume "0 < e"
with \<open>f x \<noteq> z\<close> 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 \<open>x\<in>space M\<close> \<open>f x \<noteq> z\<close> 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 \ B n i" by auto
show "eventually (\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 \ j"
with i have "dist (f x) (F j x) < 1 / Suc (m j x)"
by (intro *[OF _ _ i]) auto
also have "\ \ 1 / Suc n"
using j m_upper[OF _ _ i]
by (auto simp: field_simps)
also note \<open>1 / Suc n < e\<close>
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 \ A n m"
then have "dist (e m) (f x) + dist (f x) z \ 2 * dist (f x) z"
unfolding A_def by (auto simp: dist_commute)
also have "dist (e m) z \ dist (e m) (f x) + dist (f x) z"
by (rule dist_triangle)
finally (xtrans) have "dist (e m) z \ 2 * dist (f x) z" . }
then show "dist (F i x) z \ 2 * dist (f x) z"
unfolding F_def
apply auto
apply (rule LeastI2)
apply auto
done
qed
qed
lemma
fixes f :: "'a \ 'b::semiring_1" assumes "finite A"
shows sum_mult_indicator[simp]: "(\x \ A. f x * indicator (B x) (g x)) = (\x\{x\A. g x \ B x}. f x)"
and sum_indicator_mult[simp]: "(\x \ A. indicator (B x) (g x) * f x) = (\x\{x\A. g x \ B x}. f x)"
unfolding indicator_def
using assms by (auto intro!: sum.mono_neutral_cong_right split: if_split_asm)
lemma borel_measurable_induct_real[consumes 2, case_names set mult add seq]:
fixes P :: "('a \ real) \ bool"
assumes u: "u \ borel_measurable M" "\x. 0 \ u x"
assumes set: "\A. A \ sets M \ P (indicator A)"
assumes mult: "\u c. 0 \ c \ u \ borel_measurable M \ (\x. 0 \ u x) \ P u \ P (\x. c * u x)"
assumes add: "\u v. u \ borel_measurable M \ (\x. 0 \ u x) \ P u \ v \ borel_measurable M \ (\x. 0 \ v x) \ (\x. x \ space M \ u x = 0 \ v x = 0) \ P v \ P (\x. v x + u x)"
assumes seq: "\U. (\i. U i \ borel_measurable M) \ (\i x. 0 \ U i x) \ (\i. P (U i)) \ incseq U \ (\x. x \ space M \ (\i. U i x) \ u x) \ P u"
shows "P u"
proof -
have "(\x. ennreal (u x)) \ borel_measurable M" using u by auto
from borel_measurable_implies_simple_function_sequence'[OF this]
obtain U where U: "\i. simple_function M (U i)" "incseq U" "\i x. U i x < top" and
sup: "\x. (SUP i. U i x) = ennreal (u x)"
by blast
define U' where [abs_def]: "U' i x = indicator (space M) x * enn2real (U i x)" for i x
then have U'_sf[measurable]: "\i. simple_function M (U' i)"
using U by (auto intro!: simple_function_compose1[where g=enn2real])
show "P u"
proof (rule seq)
show U': "U' i \<in> borel_measurable M" "\<And>x. 0 \<le> U' i x" for i
using U by (auto
intro: borel_measurable_simple_function
intro!: borel_measurable_enn2real borel_measurable_times
simp: U'_def zero_le_mult_iff)
show "incseq U'"
using U(2,3)
by (auto simp: incseq_def le_fun_def image_iff eq_commute U'_def indicator_def enn2real_mono)
fix x assume x: "x \ space M"
have "(\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 "(\i. U i x) = (\i. ennreal (U' i x))"
using x U(3) by (auto simp: fun_eq_iff U'_def image_iff eq_commute)
moreover have "(SUP i. U i x) = ennreal (u x)"
using sup u(2) by (simp add: max_def)
ultimately show "(\i. U' i x) \ u x"
using u U' by simp
next
fix i
have "U' i ` space M \ enn2real ` (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 "\z. {y. U' i z = y \ y \ U' i ` space M \ z \ space M} = (if z \ space M then {U' i z} else {})"
by auto
ultimately have U': "(\z. \y\U' i`space M. y * indicator {x\space M. U' i x = y} z) = U' i"
by (simp add: U'_def fun_eq_iff)
have "\x. x \ U' i ` space M \ 0 \ x"
by (auto simp: U'_def)
with fin have "P (\z. \y\U' i`space M. y * indicator {x\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)
from insert.prems have nonneg: "x \ 0" "\y. y \ F \ y \ 0"
by simp_all
hence *: "P (\xa. x * indicat_real {x' \ space M. U' i x' = x} xa)"
by (intro mult set) auto
have "P (\z. x * indicat_real {x' \ space M. U' i x' = x} z +
(\<Sum>y\<in>F. y * indicat_real {x \<in> space M. U' i x = y} z))"
using insert(1-3)
by (intro add * sum_nonneg mult_nonneg_nonneg)
(auto simp: nonneg indicator_def sum_nonneg_eq_0_iff)
thus ?case
using insert.hyps by (subst sum.insert) auto
qed
with U' show "P (U' i)" by simp
qed
qed
lemma scaleR_cong_right:
fixes x :: "'a :: real_vector"
shows "(x \ 0 \ r = p) \ r *\<^sub>R x = p *\<^sub>R x"
by (cases "x = 0") auto
inductive simple_bochner_integrable :: "'a measure \ ('a \ 'b::real_vector) \ bool" for M f where
"simple_function M f \ emeasure M {y\space M. f y \ 0} \ \ \
simple_bochner_integrable M f"
lemma simple_bochner_integrable_compose2:
assumes p_0: "p 0 0 = 0"
shows "simple_bochner_integrable M f \ simple_bochner_integrable M g \
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 (\x. p (f x) (g x))"
by (rule simple_function_compose2)
from sf have [measurable]:
"f \ measurable M (count_space UNIV)"
"g \ measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function)
assume fin: "emeasure M {y \ space M. f y \ 0} \ \" "emeasure M {y \ space M. g y \ 0} \ \"
have "emeasure M {x\space M. p (f x) (g x) \ 0} \
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 "\ \ emeasure M {x\space M. f x \ 0} + emeasure M {x\space M. g x \ 0}"
by (intro emeasure_subadditive) auto
finally show "emeasure M {y \ space M. p (f y) (g y) \ 0} \ \"
using fin by (auto simp: top_unique)
qed
lemma simple_function_finite_support:
assumes f: "simple_function M f" and fin: "(\\<^sup>+x. f x \M) < \" and nn: "\x. 0 \ f x"
shows "emeasure M {x\space M. f x \ 0} \ \"
proof cases
from f have meas[measurable]: "f \ borel_measurable M"
by (rule borel_measurable_simple_function)
assume non_empty: "\x\space M. f x \ 0"
define m where "m = Min (f`space M - {0})"
have "m \ 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\space M. 0 \ 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 "\ \ (\\<^sup>+x. f x \M)"
using AE_space
proof (intro nn_integral_mono_AE, eventually_elim)
fix x assume "x \ space M"
with nn show "m * indicator {x \ space M. 0 \ f x} x \ f x"
using f by (auto split: split_indicator simp: simple_function_def m_def)
qed
also note \<open>\<dots> < \<infinity>\<close>
finally show ?thesis
using m by (auto simp: ennreal_mult_less_top)
next
assume "\ (\x\space M. f x \ 0)"
with nn have *: "{x\space M. f x \ 0} = {}"
by auto
show ?thesis unfolding * by simp
qed
lemma simple_bochner_integrableI_bounded:
assumes f: "simple_function M f" and fin: "(\\<^sup>+x. norm (f x) \M) < \"
shows "simple_bochner_integrable M f"
proof
have "emeasure M {y \ space M. ennreal (norm (f y)) \ 0} \ \"
proof (rule simple_function_finite_support)
show "simple_function M (\x. ennreal (norm (f x)))"
using f by (rule simple_function_compose1)
show "(\\<^sup>+ y. ennreal (norm (f y)) \M) < \" by fact
qed simp
then show "emeasure M {y \ space M. f y \ 0} \ \" by simp
qed fact
definition\<^marker>\<open>tag important\<close> simple_bochner_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> 'b::real_vector) \<Rightarrow> 'b" where
"simple_bochner_integral M f = (\y\f`space M. measure M {x\space M. f x = y} *\<^sub>R y)"
proposition simple_bochner_integral_partition:
assumes f: "simple_bochner_integrable M f" and g: "simple_function M g"
assumes sub: "\x y. x \ space M \ y \ space M \ g x = g y \ f x = f y"
assumes v: "\x. x \ space M \ f x = v (g x)"
shows "simple_bochner_integral M f = (\y\g ` space M. measure M {x\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 \ measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
from g have [measurable]: "g \ measurable M (count_space UNIV)"
by (auto intro: measurable_simple_function elim: simple_bochner_integrable.cases)
{ fix y assume "y \ space M"
then have "f ` space M \ {i. \x\space M. i = f x \ 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!: sum.cong scaleR_cong_right)
fix y assume y: "y \ space M" "f y \ 0"
have [simp]: "g ` space M \ {z. \x\space M. f y = f x \ z = g x} =
{z. \<exists>x\<in>space M. f y = f x \<and> z = g x}"
by auto
have eq:"{x \ 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. \x\space M. f y = f x \ z = g x}"
by (rule rev_finite_subset) auto
moreover
{ fix x assume "x \ space M" "f x = f y"
then have "x \ space M" "f x \ 0"
using y by auto
then have "emeasure M {y \ space M. g y = g x} \ emeasure M {y \ space M. f y \ 0}"
by (auto intro!: emeasure_mono cong: sub)
then have "emeasure M {xa \ space M. g xa = g x} < \"
using f by (auto simp: simple_bochner_integrable.simps less_top) }
ultimately
show "measure M {x \ 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: sum.If_cases eq)
apply (subst measure_finite_Union[symmetric])
apply (auto simp: disjoint_family_on_def less_top)
done
qed
also have "\ = (\y\f`space M. (\z\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!: sum.cong simp: scaleR_sum_left)
also have "\ = ?r"
by (subst sum.swap)
(auto intro!: sum.cong simp: sum.If_cases scaleR_sum_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 (\x. f x + g x) =
simple_bochner_integral M f + simple_bochner_integral M g"
proof -
from f g have "simple_bochner_integral M (\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: sum.distrib[symmetric] scaleR_add_right)
qed
lemma simple_bochner_integral_linear:
assumes "linear f"
assumes g: "simple_bochner_integrable M g"
shows "simple_bochner_integral M (\x. f (g x)) = f (simple_bochner_integral M g)"
proof -
interpret linear f by fact
from g have "simple_bochner_integral M (\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="\x y. f x"]
elim: simple_bochner_integrable.cases)
also have "\ = f (simple_bochner_integral M g)"
by (simp add: simple_bochner_integral_def sum scale)
finally show ?thesis .
qed
lemma simple_bochner_integral_minus:
assumes f: "simple_bochner_integrable M f"
shows "simple_bochner_integral M (\x. - f x) = - simple_bochner_integral M f"
proof -
from linear_uminus 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 (\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="\x y. - y"])
lemma simple_bochner_integral_norm_bound:
assumes f: "simple_bochner_integrable M f"
shows "norm (simple_bochner_integral M f) \ simple_bochner_integral M (\x. norm (f x))"
proof -
have "norm (simple_bochner_integral M f) \
(\<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_sum)
also have "\ = (\y\f ` space M. measure M {x \ space M. f x = y} *\<^sub>R norm y)"
by simp
also have "\ = simple_bochner_integral M (\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_nonneg[simp]:
fixes f :: "'a \ real"
shows "(\x. 0 \ f x) \ 0 \ simple_bochner_integral M f"
by (force simp add: simple_bochner_integral_def intro: sum_nonneg)
lemma simple_bochner_integral_eq_nn_integral:
assumes f: "simple_bochner_integrable M f" "\x. 0 \ f x"
shows "simple_bochner_integral M f = (\\<^sup>+x. f x \M)"
proof -
{ fix x y z have "(x \ 0 \ y = z) \ ennreal x * y = ennreal x * z"
by (cases "x = 0") (auto simp: zero_ennreal_def[symmetric]) }
note ennreal_cong_mult = this
have [measurable]: "f \ borel_measurable M"
using f(1) by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
{ fix y assume y: "y \ space M" "f y \ 0"
have "ennreal (measure M {x \ space M. f x = f y}) = emeasure M {x \ space M. f x = f y}"
proof (rule emeasure_eq_ennreal_measure[symmetric])
have "emeasure M {x \ space M. f x = f y} \ emeasure M {x \ space M. f x \ 0}"
using y by (intro emeasure_mono) auto
with f show "emeasure M {x \ space M. f x = f y} \ top"
by (auto simp: simple_bochner_integrable.simps top_unique)
qed
moreover have "{x \ space M. f x = f y} = (\x. ennreal (f x)) -` {ennreal (f y)} \ space M"
using f by auto
ultimately have "ennreal (measure M {x \ space M. f x = f y}) =
emeasure M ((\<lambda>x. ennreal (f x)) -` {ennreal (f y)} \<inter> space M)" by simp }
with f have "simple_bochner_integral M f = (\\<^sup>Sx. f x \M)"
unfolding simple_integral_def
by (subst simple_bochner_integral_partition[OF f(1), where g="\x. ennreal (f x)" and v=enn2real])
(auto intro: f simple_function_compose1 elim: simple_bochner_integrable.cases
intro!: sum.cong ennreal_cong_mult
simp: ac_simps ennreal_mult
simp flip: sum_ennreal)
also have "\ = (\\<^sup>+x. f x \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 \ 'b::{real_normed_vector, second_countable_topology}"
assumes f[measurable]: "f \ borel_measurable M"
assumes s: "simple_bochner_integrable M s" and t: "simple_bochner_integrable M t"
shows "ennreal (norm (simple_bochner_integral M s - simple_bochner_integral M t)) \
(\<integral>\<^sup>+ x. norm (f x - s x) \<partial>M) + (\<integral>\<^sup>+ x. norm (f x - t x) \<partial>M)"
(is "ennreal (norm (?s - ?t)) \ ?S + ?T")
proof -
have [measurable]: "s \ borel_measurable M" "t \ borel_measurable M"
using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
have "ennreal (norm (?s - ?t)) = norm (simple_bochner_integral M (\x. s x - t x))"
using s t by (subst simple_bochner_integral_diff) auto
also have "\ \ simple_bochner_integral M (\x. norm (s x - t x))"
using simple_bochner_integrable_compose2[of "(-)" M "s" "t"] s t
by (auto intro!: simple_bochner_integral_norm_bound)
also have "\ = (\\<^sup>+x. norm (s x - t x) \M)"
using simple_bochner_integrable_compose2[of "\x y. norm (x - y)" M "s" "t"] s t
by (auto intro!: simple_bochner_integral_eq_nn_integral)
also have "\ \ (\\<^sup>+x. ennreal (norm (f x - s x)) + ennreal (norm (f x - t x)) \M)"
by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(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 "\ = ?S + ?T"
by (rule nn_integral_add) auto
finally show ?thesis .
qed
inductive has_bochner_integral :: "'a measure \ ('a \ 'b) \ 'b::{real_normed_vector, second_countable_topology} \ bool"
for M f x where
"f \ borel_measurable M \
(\<And>i. simple_bochner_integrable M (s i)) \<Longrightarrow>
(\<lambda>i. \<integral>\<^sup>+x. norm (f x - s i x) \<partial>M) \<longlonglongrightarrow> 0 \<Longrightarrow>
(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x \<Longrightarrow>
has_bochner_integral M f x"
lemma has_bochner_integral_cong:
assumes "M = N" "\x. x \ space N \ f x = g x" "x = y"
shows "has_bochner_integral M f x \ has_bochner_integral N g y"
unfolding has_bochner_integral.simps assms(1,3)
using assms(2) by (simp cong: measurable_cong_simp nn_integral_cong_simp)
lemma has_bochner_integral_cong_AE:
"f \ borel_measurable M \ g \ borel_measurable M \ (AE x in M. f x = g x) \
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="\x. x \ 0"]
nn_integral_cong_AE)
auto
lemma borel_measurable_has_bochner_integral:
"has_bochner_integral M f x \ f \ borel_measurable M"
by (rule has_bochner_integral.cases)
lemma borel_measurable_has_bochner_integral'[measurable_dest]:
"has_bochner_integral M f x \ g \ measurable N M \ (\x. f (g x)) \ borel_measurable N"
using borel_measurable_has_bochner_integral[measurable] by measurable
lemma has_bochner_integral_simple_bochner_integrable:
"simple_bochner_integrable M f \ has_bochner_integral M f (simple_bochner_integral M f)"
by (rule has_bochner_integral.intros[where s="\_. f"])
(auto intro: borel_measurable_simple_function
elim: simple_bochner_integrable.cases
simp: zero_ennreal_def[symmetric])
lemma has_bochner_integral_real_indicator:
assumes [measurable]: "A \ sets M" and A: "emeasure M A < \"
shows "has_bochner_integral M (indicator A) (measure M A)"
proof -
have sbi: "simple_bochner_integrable M (indicator A::'a \ real)"
proof
have "{y \ space M. (indicator A y::real) \ 0} = A"
using sets.sets_into_space[OF \<open>A\<in>sets M\<close>] by (auto split: split_indicator)
then show "emeasure M {y \ space M. (indicator A y::real) \ 0} \ \"
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: ennreal_indicator emeasure_eq_ennreal_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 \ has_bochner_integral M g y \
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: "(\i. \\<^sup>+ x. norm (f x - sf i x) \M) \ 0"
assume g_sg: "(\i. \\<^sup>+ x. norm (g x - sg i x) \M) \ 0"
assume sf: "\i. simple_bochner_integrable M (sf i)"
and sg: "\i. simple_bochner_integrable M (sg i)"
then have [measurable]: "\i. sf i \ borel_measurable M" "\i. sg i \ borel_measurable M"
by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
assume [measurable]: "f \ borel_measurable M" "g \ borel_measurable M"
show "\i. simple_bochner_integrable M (\x. sf i x + sg i x)"
using sf sg by (simp add: simple_bochner_integrable_compose2)
show "(\i. \\<^sup>+ x. (norm (f x + g x - (sf i x + sg i x))) \M) \ 0"
(is "?f \ 0")
proof (rule tendsto_sandwich)
show "eventually (\n. 0 \ ?f n) sequentially" "(\_. 0) \ 0"
by auto
show "eventually (\i. ?f i \ (\\<^sup>+ x. (norm (f x - sf i x)) \M) + \\<^sup>+ x. (norm (g x - sg i x)) \M) sequentially"
(is "eventually (\i. ?f i \ ?g i) sequentially")
proof (intro always_eventually allI)
fix i have "?f i \ (\\<^sup>+ x. (norm (f x - sf i x)) + ennreal (norm (g x - sg i x)) \M)"
by (auto intro!: nn_integral_mono norm_diff_triangle_ineq
simp flip: ennreal_plus)
also have "\ = ?g i"
by (intro nn_integral_add) auto
finally show "?f i \ ?g i" .
qed
show "?g \ 0"
using tendsto_add[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 \ has_bochner_integral M (\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 \ borel_measurable borel"
by (intro borel_measurable_continuous_onI T.continuous_on continuous_on_id)
assume [measurable]: "f \ borel_measurable M"
then show "(\x. T (f x)) \ borel_measurable M"
by auto
fix s assume f_s: "(\i. \\<^sup>+ x. norm (f x - s i x) \M) \ 0"
assume s: "\i. simple_bochner_integrable M (s i)"
then show "\i. simple_bochner_integrable M (\x. T (s i x))"
by (auto intro: simple_bochner_integrable_compose2 T.zero)
have [measurable]: "\i. s i \ borel_measurable M"
using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
obtain K where K: "K > 0" "\x i. norm (T (f x) - T (s i x)) \ norm (f x - s i x) * K"
using T.pos_bounded by (auto simp: T.diff[symmetric])
show "(\i. \\<^sup>+ x. norm (T (f x) - T (s i x)) \M) \ 0"
(is "?f \ 0")
proof (rule tendsto_sandwich)
show "eventually (\n. 0 \ ?f n) sequentially" "(\_. 0) \ 0"
by auto
show "eventually (\i. ?f i \ K * (\\<^sup>+ x. norm (f x - s i x) \M)) sequentially"
(is "eventually (\i. ?f i \ ?g i) sequentially")
proof (intro always_eventually allI)
fix i have "?f i \ (\\<^sup>+ x. ennreal K * norm (f x - s i x) \M)"
using K by (intro nn_integral_mono) (auto simp: ac_simps ennreal_mult[symmetric])
also have "\ = ?g i"
using K by (intro nn_integral_cmult) auto
finally show "?f i \ ?g i" .
qed
show "?g \ 0"
using ennreal_tendsto_cmult[OF _ f_s] by simp
qed
assume "(\i. simple_bochner_integral M (s i)) \ x"
with s show "(\i. simple_bochner_integral M (\x. T (s i x))) \ T x"
by (auto intro!: T.tendsto simp: simple_bochner_integral_linear T.linear_axioms)
qed
lemma has_bochner_integral_zero[intro]: "has_bochner_integral M (\x. 0) 0"
by (auto intro!: has_bochner_integral.intros[where s="\_ _. 0"]
simp: zero_ennreal_def[symmetric] simple_bochner_integrable.simps
simple_bochner_integral_def image_constant_conv)
lemma has_bochner_integral_scaleR_left[intro]:
"(c \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\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 \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\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 \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\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 \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\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, second_countable_topology}"
shows "(c \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\x. f x / c) (x / c)"
using has_bochner_integral_divide by (cases "c = 0") auto
lemma has_bochner_integral_inner_left[intro]:
"(c \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\x. f x \ c) (x \ 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 \ 0 \ has_bochner_integral M f x) \ has_bochner_integral M (\x. c \ f x) (c \ 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 \ sets M \ emeasure M A < \ \
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 \ has_bochner_integral M g y \
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_sum:
"(\i. i \ I \ has_bochner_integral M (f i) (x i)) \
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
proposition has_bochner_integral_implies_finite_norm:
"has_bochner_integral M f x \ (\\<^sup>+x. norm (f x) \M) < \"
proof (elim has_bochner_integral.cases)
fix s v
assume [measurable]: "f \ borel_measurable M" and s: "\i. simple_bochner_integrable M (s i)" and
lim_0: "(\i. \\<^sup>+ x. ennreal (norm (f x - s i x)) \M) \ 0"
from order_tendstoD[OF lim_0, of "\"]
obtain i where f_s_fin: "(\\<^sup>+ x. ennreal (norm (f x - s i x)) \M) < \"
by (auto simp: eventually_sequentially)
have [measurable]: "\i. s i \ borel_measurable M"
using s by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
define m where "m = (if space M = {} then 0 else Max ((\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 "\x. x \ space M \ norm (s i x) \ m" "0 \ m"
by (auto simp: m_def image_comp comp_def Max_ge_iff)
then have "(\\<^sup>+x. norm (s i x) \M) \ (\\<^sup>+x. ennreal m * indicator {x\space M. s i x \ 0} x \M)"
by (auto split: split_indicator intro!: Max_ge nn_integral_mono simp:)
also have "\ < \"
using s by (subst nn_integral_cmult_indicator) (auto simp: \<open>0 \<le> m\<close> simple_bochner_integrable.simps ennreal_mult_less_top less_top)
finally have s_fin: "(\\<^sup>+x. norm (s i x) \M) < \" .
have "(\\<^sup>+ x. norm (f x) \M) \ (\\<^sup>+ x. ennreal (norm (f x - s i x)) + ennreal (norm (s i x)) \M)"
by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(metis add.commute norm_triangle_sub)
also have "\ = (\\<^sup>+x. norm (f x - s i x) \M) + (\\<^sup>+x. norm (s i x) \M)"
by (rule nn_integral_add) auto
also have "\ < \"
using s_fin f_s_fin by auto
finally show "(\\<^sup>+ x. ennreal (norm (f x)) \M) < \" .
qed
proposition has_bochner_integral_norm_bound:
assumes i: "has_bochner_integral M f x"
shows "norm x \ (\\<^sup>+x. norm (f x) \M)"
using assms proof
fix s assume
x: "(\i. simple_bochner_integral M (s i)) \ x" (is "?s \ x") and
s[simp]: "\i. simple_bochner_integrable M (s i)" and
lim: "(\i. \\<^sup>+ x. ennreal (norm (f x - s i x)) \M) \ 0" and
f[measurable]: "f \ borel_measurable M"
have [measurable]: "\i. s i \ borel_measurable M"
using s by (auto simp: simple_bochner_integrable.simps intro: borel_measurable_simple_function)
show "norm x \ (\\<^sup>+x. norm (f x) \M)"
proof (rule LIMSEQ_le)
show "(\i. ennreal (norm (?s i))) \ norm x"
using x by (auto simp: tendsto_ennreal_iff intro: tendsto_intros)
show "\N. \n\N. norm (?s n) \ (\\<^sup>+x. norm (f x - s n x) \M) + (\\<^sup>+x. norm (f x) \M)"
(is "\N. \n\N. _ \ ?t n")
proof (intro exI allI impI)
fix n
have "ennreal (norm (?s n)) \ simple_bochner_integral M (\x. norm (s n x))"
by (auto intro!: simple_bochner_integral_norm_bound)
also have "\ = (\\<^sup>+x. norm (s n x) \M)"
by (intro simple_bochner_integral_eq_nn_integral)
(auto intro: s simple_bochner_integrable_compose2)
also have "\ \ (\\<^sup>+x. ennreal (norm (f x - s n x)) + norm (f x) \M)"
by (auto intro!: nn_integral_mono simp flip: ennreal_plus)
(metis add.commute norm_minus_commute norm_triangle_sub)
also have "\ = ?t n"
by (rule nn_integral_add) auto
finally show "norm (?s n) \ ?t n" .
qed
have "?t \ 0 + (\\<^sup>+ x. ennreal (norm (f x)) \M)"
using has_bochner_integral_implies_finite_norm[OF i]
by (intro tendsto_add tendsto_const lim)
then show "?t \ \\<^sup>+ x. ennreal (norm (f x)) \M"
by simp
qed
qed
lemma has_bochner_integral_eq:
"has_bochner_integral M f x \ has_bochner_integral M f y \ x = y"
proof (elim has_bochner_integral.cases)
assume f[measurable]: "f \ borel_measurable M"
fix s t
assume "(\i. \\<^sup>+ x. norm (f x - s i x) \M) \ 0" (is "?S \ 0")
assume "(\i. \\<^sup>+ x. norm (f x - t i x) \M) \ 0" (is "?T \ 0")
assume s: "\i. simple_bochner_integrable M (s i)"
assume t: "\i. simple_bochner_integrable M (t i)"
have [measurable]: "\i. s i \ borel_measurable M" "\i. t i \ borel_measurable M"
using s t by (auto intro: borel_measurable_simple_function elim: simple_bochner_integrable.cases)
let ?s = "\i. simple_bochner_integral M (s i)"
let ?t = "\i. simple_bochner_integral M (t i)"
assume "?s \ x" "?t \ y"
then have "(\i. norm (?s i - ?t i)) \ norm (x - y)"
by (intro tendsto_intros)
moreover
have "(\i. ennreal (norm (?s i - ?t i))) \ ennreal 0"
proof (rule tendsto_sandwich)
show "eventually (\i. 0 \ ennreal (norm (?s i - ?t i))) sequentially" "(\_. 0) \ ennreal 0"
by auto
show "eventually (\i. norm (?s i - ?t i) \ ?S i + ?T i) sequentially"
by (intro always_eventually allI simple_bochner_integral_bounded s t f)
show "(\i. ?S i + ?T i) \ ennreal 0"
using tendsto_add[OF \<open>?S \<longlonglongrightarrow> 0\<close> \<open>?T \<longlonglongrightarrow> 0\<close>] by simp
qed
then have "(\i. norm (?s i - ?t i)) \ 0"
by (simp flip: ennreal_0)
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 \ 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 "(\i. \\<^sup>+ x. ennreal (norm (f x - s i x)) \M) \ 0"
also have "(\i. \\<^sup>+ x. ennreal (norm (f x - s i x)) \M) = (\i. \\<^sup>+ x. ennreal (norm (g x - s i x)) \M)"
using ae
by (intro ext nn_integral_cong_AE, eventually_elim) simp
finally show "(\i. \\<^sup>+ x. ennreal (norm (g x - s i x)) \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 :: "_ \ 'b::real_normed_vector"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
shows "simple_bochner_integrable (restrict_space M \) f \
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_left_commute)
lemma simple_bochner_integral_restrict_space:
fixes f :: "_ \ 'b::real_normed_vector"
assumes \<Omega>: "\<Omega> \<inter> space M \<in> sets M"
assumes f: "simple_bochner_integrable (restrict_space M \) f"
shows "simple_bochner_integral (restrict_space M \) f =
simple_bochner_integral M (\<lambda>x. indicator \<Omega> x *\<^sub>R f x)"
proof -
have "finite ((\x. indicator \ 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!: sum.mono_neutral_cong_left arg_cong2[where f=measure])
qed
context
notes [[inductive_internals]]
begin
inductive integrable for M f where
"has_bochner_integral M f x \ integrable M f"
end
definition\<^marker>\<open>tag important\<close> lebesgue_integral ("integral\<^sup>L") where
"integral\<^sup>L M f = (if \x. has_bochner_integral M f x then THE x. has_bochner_integral M f x else 0)"
syntax
"_lebesgue_integral" :: "pttrn \ real \ 'a measure \ real" ("\((2 _./ _)/ \_)" [60,61] 110)
translations
"\ x. f \M" == "CONST lebesgue_integral M (\x. f)"
syntax
"_ascii_lebesgue_integral" :: "pttrn \ 'a measure \ real \ real" ("(3LINT (1_)/|(_)./ _)" [0,110,60] 60)
translations
"LINT x|M. f" == "CONST lebesgue_integral M (\x. f)"
lemma has_bochner_integral_integral_eq: "has_bochner_integral M f x \ 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 \ 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 \ integrable M f \ 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 \ 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: "\ integrable M f \ integral\<^sup>L M f = 0"
unfolding integrable.simps lebesgue_integral_def by (auto intro!: arg_cong[where f=The])
lemma integral_eq_cases:
"integrable M f \ integrable N g \
(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 \ f \ borel_measurable M"
by (auto elim: integrable.cases has_bochner_integral.cases)
lemma borel_measurable_integrable'[measurable_dest]:
"integrable M f \ g \ measurable N M \ (\x. f (g x)) \ borel_measurable N"
using borel_measurable_integrable[measurable] by measurable
lemma integrable_cong:
"M = N \ (\x. x \ space N \ f x = g x) \ integrable M f \ integrable N g"
by (simp cong: has_bochner_integral_cong add: integrable.simps)
lemma integrable_cong_AE:
"f \ borel_measurable M \ g \ borel_measurable M \ AE x in M. f x = g x \
integrable M f \<longleftrightarrow> integrable M g"
unfolding integrable.simps
by (intro has_bochner_integral_cong_AE arg_cong[where f=Ex] ext)
lemma integrable_cong_AE_imp:
"integrable M g \ f \ borel_measurable M \ (AE x in M. g x = f x) \ integrable M f"
using integrable_cong_AE[of f M g] by (auto simp: eq_commute)
lemma integral_cong:
"M = N \ (\x. x \ space N \ f x = g x) \ integral\<^sup>L M f = integral\<^sup>L N g"
by (simp cong: has_bochner_integral_cong cong del: if_weak_cong add: lebesgue_integral_def)
lemma integral_cong_AE:
"f \ borel_measurable M \ g \ borel_measurable M \ AE x in M. f x = g x \
integral\<^sup>L M f = integral\<^sup>L M g"
unfolding lebesgue_integral_def
by (rule arg_cong[where x="has_bochner_integral M f"]) (intro has_bochner_integral_cong_AE ext)
lemma integrable_add[simp, intro]: "integrable M f \ integrable M g \ integrable M (\x. f x + g x)"
by (auto simp: integrable.simps)
lemma integrable_zero[simp, intro]: "integrable M (\x. 0)"
by (metis has_bochner_integral_zero integrable.simps)
lemma integrable_sum[simp, intro]: "(\i. i \ I \ integrable M (f i)) \ integrable M (\x. \i\I. f i x)"
by (metis has_bochner_integral_sum integrable.simps)
lemma integrable_indicator[simp, intro]: "A \ sets M \ emeasure M A < \ \
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 \ sets M \ emeasure M A < \ \
integrable M (indicator A :: 'a \ real)"
by (metis has_bochner_integral_real_indicator integrable.simps)
lemma integrable_diff[simp, intro]: "integrable M f \ integrable M g \ integrable M (\x. f x - g x)"
by (auto simp: integrable.simps intro: has_bochner_integral_diff)
lemma integrable_bounded_linear: "bounded_linear T \ integrable M f \ integrable M (\x. T (f x))"
by (auto simp: integrable.simps intro: has_bochner_integral_bounded_linear)
lemma integrable_scaleR_left[simp, intro]: "(c \ 0 \ integrable M f) \ integrable M (\x. f x *\<^sub>R c)"
unfolding integrable.simps by fastforce
lemma integrable_scaleR_right[simp, intro]: "(c \ 0 \ integrable M f) \ integrable M (\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 \ 0 \ integrable M f) \ integrable M (\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 \ 0 \ integrable M f) \ integrable M (\x. c * f x)"
unfolding integrable.simps by fastforce
lemma integrable_divide_zero[simp, intro]:
fixes c :: "_::{real_normed_field, field, second_countable_topology}"
shows "(c \ 0 \ integrable M f) \ integrable M (\x. f x / c)"
unfolding integrable.simps by fastforce
lemma integrable_inner_left[simp, intro]:
"(c \ 0 \ integrable M f) \ integrable M (\x. f x \ c)"
unfolding integrable.simps by fastforce
lemma integrable_inner_right[simp, intro]:
"(c \ 0 \ integrable M f) \ integrable M (\x. c \ 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 (\x. 0) = 0"
by (intro has_bochner_integral_integral_eq has_bochner_integral_zero)
lemma integral_add[simp]: "integrable M f \ integrable M g \
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 \ integrable M g \
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_sum: "(\i. i \ I \ integrable M (f i)) \
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_sum has_bochner_integral_integrable)
lemma integral_sum'[simp]: "(\i. i \ I =simp=> integrable M (f i)) \
integral\<^sup>L M (\<lambda>x. \<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
unfolding simp_implies_def by (rule integral_sum)
lemma integral_bounded_linear: "bounded_linear T \ integrable M f \
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_bounded_linear':
assumes T: "bounded_linear T" and T': "bounded_linear T'"
assumes *: "\ (\x. T x = 0) \ (\x. T' (T x) = x)"
shows "integral\<^sup>L M (\x. T (f x)) = T (integral\<^sup>L M f)"
proof cases
assume "(\x. T x = 0)" then show ?thesis
by simp
next
assume **: "\ (\x. T x = 0)"
show ?thesis
proof cases
assume "integrable M f" with T show ?thesis
by (rule integral_bounded_linear)
next
assume not: "\ integrable M f"
moreover have "\ integrable M (\x. T (f x))"
proof
assume "integrable M (\x. T (f x))"
from integrable_bounded_linear[OF T' this] not *[OF **]
show False
by auto
qed
ultimately show ?thesis
using T by (simp add: not_integrable_integral_eq linear_simps)
qed
qed
lemma integral_scaleR_left[simp]: "(c \ 0 \ integrable M f) \ (\ x. f x *\<^sub>R c \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]: "(\ x. c *\<^sub>R f x \M) = c *\<^sub>R integral\<^sup>L M f"
by (rule integral_bounded_linear'[OF bounded_linear_scaleR_right bounded_linear_scaleR_right[of "1 / c"]]) simp
lemma integral_mult_left[simp]:
fixes c :: "_::{real_normed_algebra,second_countable_topology}"
shows "(c \ 0 \ integrable M f) \ (\ x. f x * c \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 \ 0 \ integrable M f) \ (\ x. c * f x \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_mult_left_zero[simp]:
fixes c :: "_::{real_normed_field,second_countable_topology}"
shows "(\ x. f x * c \M) = integral\<^sup>L M f * c"
by (rule integral_bounded_linear'[OF bounded_linear_mult_left bounded_linear_mult_left[of "1 / c"]]) simp
lemma integral_mult_right_zero[simp]:
fixes c :: "_::{real_normed_field,second_countable_topology}"
shows "(\ x. c * f x \M) = c * integral\<^sup>L M f"
by (rule integral_bounded_linear'[OF bounded_linear_mult_right bounded_linear_mult_right[of "1 / c"]]) simp
lemma integral_inner_left[simp]: "(c \ 0 \ integrable M f) \ (\ x. f x \ c \M) = integral\<^sup>L M f \ c"
by (intro has_bochner_integral_integral_eq has_bochner_integral_integrable has_bochner_integral_inner_left)
lemma integral_inner_right[simp]: "(c \ 0 \ integrable M f) \ (\ x. c \ f x \M) = c \ 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, second_countable_topology}"
shows "integral\<^sup>L M (\x. f x / c) = integral\<^sup>L M f / c"
by (rule integral_bounded_linear'[OF bounded_linear_divide bounded_linear_mult_left[of c]]) simp
lemma integral_minus[simp]: "integral\<^sup>L M (\x. - f x) = - integral\<^sup>L M f"
by (rule integral_bounded_linear'[OF bounded_linear_minus[OF bounded_linear_ident] bounded_linear_minus[OF bounded_linear_ident]]) simp
lemma integral_complex_of_real[simp]: "integral\<^sup>L M (\x. complex_of_real (f x)) = of_real (integral\<^sup>L M f)"
by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_Re]) simp
lemma integral_cnj[simp]: "integral\<^sup>L M (\x. cnj (f x)) = cnj (integral\<^sup>L M f)"
by (rule integral_bounded_linear'[OF bounded_linear_cnj bounded_linear_cnj]) simp
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_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_ennreal:
"integrable M f \ norm (integral\<^sup>L M f) \ (\\<^sup>+x. norm (f x) \M)"
by (metis has_bochner_integral_integrable has_bochner_integral_norm_bound)
lemma integrableI_sequence:
fixes f :: "'a \ 'b::{banach, second_countable_topology}"
assumes f[measurable]: "f \ borel_measurable M"
assumes s: "\i. simple_bochner_integrable M (s i)"
assumes lim: "(\i. \\<^sup>+x. norm (f x - s i x) \M) \ 0" (is "?S \ 0")
shows "integrable M f"
proof -
let ?s = "\n. simple_bochner_integral M (s n)"
have "\x. ?s \ x"
unfolding convergent_eq_Cauchy
proof (rule metric_CauchyI)
fix e :: real assume "0 < e"
then have "0 < ennreal (e / 2)" by auto
from order_tendstoD(2)[OF lim this]
obtain M where M: "\n. M \ n \ ?S n < e / 2"
by (auto simp: eventually_sequentially)
show "\M. \m\M. \n\M. dist (?s m) (?s n) < e"
proof (intro exI allI impI)
fix m n assume m: "M \ m" and n: "M \ n"
have "?S n \ \"
using M[OF n] by auto
have "norm (?s n - ?s m) \ ?S n + ?S m"
by (intro simple_bochner_integral_bounded s f)
also have "\ < ennreal (e / 2) + e / 2"
by (intro add_strict_mono M n m)
also have "\ = e" using \0 by (simp flip: ennreal_plus)
finally show "dist (?s n) (?s m) < e"
using \<open>0<e\<close> by (simp add: dist_norm ennreal_less_iff)
qed
qed
then obtain x where "?s \ x" ..
show ?thesis
by (rule, rule) fact+
qed
proposition nn_integral_dominated_convergence_norm:
fixes u' :: "_ \ _::{real_normed_vector, second_countable_topology}"
assumes [measurable]:
"\i. u i \ borel_measurable M" "u' \ borel_measurable M" "w \ borel_measurable M"
and bound: "\j. AE x in M. norm (u j x) \ w x"
and w: "(\\<^sup>+x. w x \M) < \"
and u': "AE x in M. (\i. u i x) \ u' x"
shows "(\i. (\\<^sup>+x. norm (u' x - u i x) \M)) \ 0"
proof -
have "AE x in M. \j. norm (u j x) \ w x"
unfolding AE_all_countable by rule fact
with u' have bnd: "AE x in M. \j. norm (u' x - u j x) \ 2 * w x"
proof (eventually_elim, intro allI)
fix i x assume "(\i. u i x) \ u' x" "\j. norm (u j x) \ w x" "\j. norm (u j x) \ w x"
then have "norm (u' x) \ w x" "norm (u i x) \ w x"
by (auto intro: LIMSEQ_le_const2 tendsto_norm)
then have "norm (u' x) + norm (u i x) \ 2 * w x"
by simp
also have "norm (u' x - u i x) \ norm (u' x) + norm (u i x)"
by (rule norm_triangle_ineq4)
finally (xtrans) show "norm (u' x - u i x) \ 2 * w x" .
qed
have w_nonneg: "AE x in M. 0 \ w x"
using bound[of 0] by (auto intro: order_trans[OF norm_ge_zero])
have "(\i. (\\<^sup>+x. norm (u' x - u i x) \M)) \ (\\<^sup>+x. 0 \M)"
proof (rule nn_integral_dominated_convergence)
show "(\\<^sup>+x. 2 * w x \M) < \"
by (rule nn_integral_mult_bounded_inf[OF _ w, of 2]) (insert w_nonneg, auto simp: ennreal_mult )
show "AE x in M. (\i. ennreal (norm (u' x - u i x))) \ 0"
using u'
proof eventually_elim
fix x assume "(\i. u i x) \ u' x"
from tendsto_diff[OF tendsto_const[of "u' x"] this]
show "(\i. ennreal (norm (u' x - u i x))) \ 0"
by (simp add: tendsto_norm_zero_iff flip: ennreal_0)
qed
qed (insert bnd w_nonneg, auto)
then show ?thesis by simp
qed
proposition integrableI_bounded:
fixes f :: "'a \ 'b::{banach, second_countable_topology}"
assumes f[measurable]: "f \ borel_measurable M" and fin: "(\\<^sup>+x. norm (f x) \M) < \"
shows "integrable M f"
proof -
from borel_measurable_implies_sequence_metric[OF f, of 0] obtain s where
s: "\i. simple_function M (s i)" and
pointwise: "\x. x \ space M \ (\i. s i x) \ f x" and
--> --------------------
--> maximum size reached
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