move absolutely_integrable_on to Equivalence_Lebesgue_Henstock_Integration, now based on the Lebesgue integral
(* Title: HOL/Analysis/Set_Integral.thy
Author: Jeremy Avigad (CMU), Johannes Hölzl (TUM), Luke Serafin (CMU)
Notation and useful facts for working with integrals over a set.
TODO: keep all these? Need unicode translations as well.
*)
theory Set_Integral
imports Bochner_Integration
begin
(*
Notation
*)
abbreviation "set_borel_measurable M A f \<equiv> (\<lambda>x. indicator A x *\<^sub>R f x) \<in> borel_measurable M"
abbreviation "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicator A x *\<^sub>R f x)"
abbreviation "set_lebesgue_integral M A f \<equiv> lebesgue_integral M (\<lambda>x. indicator A x *\<^sub>R f x)"
syntax
"_ascii_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
("(4LINT (_):(_)/|(_)./ _)" [0,60,110,61] 60)
translations
"LINT x:A|M. f" == "CONST set_lebesgue_integral M A (\<lambda>x. f)"
abbreviation
"set_almost_everywhere A M P \<equiv> AE x in M. x \<in> A \<longrightarrow> P x"
syntax
"_set_almost_everywhere" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool \<Rightarrow> bool"
("AE _\<in>_ in _./ _" [0,0,0,10] 10)
translations
"AE x\<in>A in M. P" == "CONST set_almost_everywhere A M (\<lambda>x. P)"
(*
Notation for integration wrt lebesgue measure on the reals:
LBINT x. f
LBINT x : A. f
TODO: keep all these? Need unicode.
*)
syntax
"_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real"
("(2LBINT _./ _)" [0,60] 60)
syntax
"_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> real"
("(3LBINT _:_./ _)" [0,60,61] 60)
(*
Basic properties
*)
(*
lemma indicator_abs_eq: "\<And>A x. \<bar>indicator A x\<bar> = ((indicator A x) :: real)"
by (auto simp add: indicator_def)
*)
lemma set_borel_measurable_sets:
fixes f :: "_ \<Rightarrow> _::real_normed_vector"
assumes "set_borel_measurable M X f" "B \<in> sets borel" "X \<in> sets M"
shows "f -` B \<inter> X \<in> sets M"
proof -
have "f \<in> borel_measurable (restrict_space M X)"
using assms by (subst borel_measurable_restrict_space_iff) auto
then have "f -` B \<inter> space (restrict_space M X) \<in> sets (restrict_space M X)"
by (rule measurable_sets) fact
with \<open>X \<in> sets M\<close> show ?thesis
by (subst (asm) sets_restrict_space_iff) (auto simp: space_restrict_space)
qed
lemma set_lebesgue_integral_cong:
assumes "A \<in> sets M" and "\<forall>x. x \<in> A \<longrightarrow> f x = g x"
shows "(LINT x:A|M. f x) = (LINT x:A|M. g x)"
using assms by (auto intro!: Bochner_Integration.integral_cong split: split_indicator simp add: sets.sets_into_space)
lemma set_lebesgue_integral_cong_AE:
assumes [measurable]: "A \<in> sets M" "f \<in> borel_measurable M" "g \<in> borel_measurable M"
assumes "AE x \<in> A in M. f x = g x"
shows "LINT x:A|M. f x = LINT x:A|M. g x"
proof-
have "AE x in M. indicator A x *\<^sub>R f x = indicator A x *\<^sub>R g x"
using assms by auto
thus ?thesis by (intro integral_cong_AE) auto
qed
lemma set_integrable_cong_AE:
"f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
AE x \<in> A in M. f x = g x \<Longrightarrow> A \<in> sets M \<Longrightarrow>
set_integrable M A f = set_integrable M A g"
by (rule integrable_cong_AE) auto
lemma set_integrable_subset:
fixes M A B and f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "set_integrable M A f" "B \<in> sets M" "B \<subseteq> A"
shows "set_integrable M B f"
proof -
have "set_integrable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
by (rule integrable_mult_indicator) fact+
with \<open>B \<subseteq> A\<close> show ?thesis
by (simp add: indicator_inter_arith[symmetric] Int_absorb2)
qed
(* TODO: integral_cmul_indicator should be named set_integral_const *)
(* TODO: borel_integrable_atLeastAtMost should be named something like set_integrable_Icc_isCont *)
lemma set_integral_scaleR_right [simp]: "LINT t:A|M. a *\<^sub>R f t = a *\<^sub>R (LINT t:A|M. f t)"
by (subst integral_scaleR_right[symmetric]) (auto intro!: Bochner_Integration.integral_cong)
lemma set_integral_mult_right [simp]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "LINT t:A|M. a * f t = a * (LINT t:A|M. f t)"
by (subst integral_mult_right_zero[symmetric]) auto
lemma set_integral_mult_left [simp]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "LINT t:A|M. f t * a = (LINT t:A|M. f t) * a"
by (subst integral_mult_left_zero[symmetric]) auto
lemma set_integral_divide_zero [simp]:
fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
shows "LINT t:A|M. f t / a = (LINT t:A|M. f t) / a"
by (subst integral_divide_zero[symmetric], intro Bochner_Integration.integral_cong)
(auto split: split_indicator)
lemma set_integrable_scaleR_right [simp, intro]:
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a *\<^sub>R f t)"
unfolding scaleR_left_commute by (rule integrable_scaleR_right)
lemma set_integrable_scaleR_left [simp, intro]:
fixes a :: "_ :: {banach, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t *\<^sub>R a)"
using integrable_scaleR_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_mult_right [simp, intro]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. a * f t)"
using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_mult_left [simp, intro]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)"
using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
lemma set_integrable_divide [simp, intro]:
fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f"
shows "set_integrable M A (\<lambda>t. f t / a)"
proof -
have "integrable M (\<lambda>x. indicator A x *\<^sub>R f x / a)"
using assms by (rule integrable_divide_zero)
also have "(\<lambda>x. indicator A x *\<^sub>R f x / a) = (\<lambda>x. indicator A x *\<^sub>R (f x / a))"
by (auto split: split_indicator)
finally show ?thesis .
qed
lemma set_integral_add [simp, intro]:
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "set_integrable M A f" "set_integrable M A g"
shows "set_integrable M A (\<lambda>x. f x + g x)"
and "LINT x:A|M. f x + g x = (LINT x:A|M. f x) + (LINT x:A|M. g x)"
using assms by (simp_all add: scaleR_add_right)
lemma set_integral_diff [simp, intro]:
assumes "set_integrable M A f" "set_integrable M A g"
shows "set_integrable M A (\<lambda>x. f x - g x)" and "LINT x:A|M. f x - g x =
(LINT x:A|M. f x) - (LINT x:A|M. g x)"
using assms by (simp_all add: scaleR_diff_right)
(* question: why do we have this for negation, but multiplication by a constant
requires an integrability assumption? *)
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)"
by (subst integral_minus[symmetric]) simp_all
lemma set_integral_complex_of_real:
"LINT x:A|M. complex_of_real (f x) = of_real (LINT x:A|M. f x)"
by (subst integral_complex_of_real[symmetric])
(auto intro!: Bochner_Integration.integral_cong split: split_indicator)
lemma set_integral_mono:
fixes f g :: "_ \<Rightarrow> real"
assumes "set_integrable M A f" "set_integrable M A g"
"\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
using assms by (auto intro: integral_mono split: split_indicator)
lemma set_integral_mono_AE:
fixes f g :: "_ \<Rightarrow> real"
assumes "set_integrable M A f" "set_integrable M A g"
"AE x \<in> A in M. f x \<le> g x"
shows "(LINT x:A|M. f x) \<le> (LINT x:A|M. g x)"
using assms by (auto intro: integral_mono_AE split: split_indicator)
lemma set_integrable_abs: "set_integrable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar> :: real)"
using integrable_abs[of M "\<lambda>x. f x * indicator A x"] by (simp add: abs_mult ac_simps)
lemma set_integrable_abs_iff:
fixes f :: "_ \<Rightarrow> real"
shows "set_borel_measurable M A f \<Longrightarrow> set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
by (subst (2) integrable_abs_iff[symmetric]) (simp_all add: abs_mult ac_simps)
lemma set_integrable_abs_iff':
fixes f :: "_ \<Rightarrow> real"
shows "f \<in> borel_measurable M \<Longrightarrow> A \<in> sets M \<Longrightarrow>
set_integrable M A (\<lambda>x. \<bar>f x\<bar>) = set_integrable M A f"
by (intro set_integrable_abs_iff) auto
lemma set_integrable_discrete_difference:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "countable X"
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
shows "set_integrable M A f \<longleftrightarrow> set_integrable M B f"
proof (rule integrable_discrete_difference[where X=X])
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
using diff by (auto split: split_indicator)
qed fact+
lemma set_integral_discrete_difference:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "countable X"
assumes diff: "(A - B) \<union> (B - A) \<subseteq> X"
assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
proof (rule integral_discrete_difference[where X=X])
show "\<And>x. x \<in> space M \<Longrightarrow> x \<notin> X \<Longrightarrow> indicator A x *\<^sub>R f x = indicator B x *\<^sub>R f x"
using diff by (auto split: split_indicator)
qed fact+
lemma set_integrable_Un:
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes f_A: "set_integrable M A f" and f_B: "set_integrable M B f"
and [measurable]: "A \<in> sets M" "B \<in> sets M"
shows "set_integrable M (A \<union> B) f"
proof -
have "set_integrable M (A - B) f"
using f_A by (rule set_integrable_subset) auto
from Bochner_Integration.integrable_add[OF this f_B] show ?thesis
by (rule integrable_cong[THEN iffD1, rotated 2]) (auto split: split_indicator)
qed
lemma set_integrable_UN:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "finite I" "\<And>i. i\<in>I \<Longrightarrow> set_integrable M (A i) f"
"\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets M"
shows "set_integrable M (\<Union>i\<in>I. A i) f"
using assms by (induct I) (auto intro!: set_integrable_Un)
lemma set_integral_Un:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "A \<inter> B = {}"
and "set_integrable M A f"
and "set_integrable M B f"
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
by (auto simp add: indicator_union_arith indicator_inter_arith[symmetric]
scaleR_add_left assms)
lemma set_integral_cong_set:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes [measurable]: "set_borel_measurable M A f" "set_borel_measurable M B f"
and ae: "AE x in M. x \<in> A \<longleftrightarrow> x \<in> B"
shows "LINT x:B|M. f x = LINT x:A|M. f x"
proof (rule integral_cong_AE)
show "AE x in M. indicator B x *\<^sub>R f x = indicator A x *\<^sub>R f x"
using ae by (auto simp: subset_eq split: split_indicator)
qed fact+
lemma set_borel_measurable_subset:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes [measurable]: "set_borel_measurable M A f" "B \<in> sets M" and "B \<subseteq> A"
shows "set_borel_measurable M B f"
proof -
have "set_borel_measurable M B (\<lambda>x. indicator A x *\<^sub>R f x)"
by measurable
also have "(\<lambda>x. indicator B x *\<^sub>R indicator A x *\<^sub>R f x) = (\<lambda>x. indicator B x *\<^sub>R f x)"
using \<open>B \<subseteq> A\<close> by (auto simp: fun_eq_iff split: split_indicator)
finally show ?thesis .
qed
lemma set_integral_Un_AE:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes ae: "AE x in M. \<not> (x \<in> A \<and> x \<in> B)" and [measurable]: "A \<in> sets M" "B \<in> sets M"
and "set_integrable M A f"
and "set_integrable M B f"
shows "LINT x:A\<union>B|M. f x = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
proof -
have f: "set_integrable M (A \<union> B) f"
by (intro set_integrable_Un assms)
then have f': "set_borel_measurable M (A \<union> B) f"
by (rule borel_measurable_integrable)
have "LINT x:A\<union>B|M. f x = LINT x:(A - A \<inter> B) \<union> (B - A \<inter> B)|M. f x"
proof (rule set_integral_cong_set)
show "AE x in M. (x \<in> A - A \<inter> B \<union> (B - A \<inter> B)) = (x \<in> A \<union> B)"
using ae by auto
show "set_borel_measurable M (A - A \<inter> B \<union> (B - A \<inter> B)) f"
using f' by (rule set_borel_measurable_subset) auto
qed fact
also have "\<dots> = (LINT x:(A - A \<inter> B)|M. f x) + (LINT x:(B - A \<inter> B)|M. f x)"
by (auto intro!: set_integral_Un set_integrable_subset[OF f])
also have "\<dots> = (LINT x:A|M. f x) + (LINT x:B|M. f x)"
using ae
by (intro arg_cong2[where f="op+"] set_integral_cong_set)
(auto intro!: set_borel_measurable_subset[OF f'])
finally show ?thesis .
qed
lemma set_integral_finite_Union:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "finite I" "disjoint_family_on A I"
and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f" "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
shows "(LINT x:(\<Union>i\<in>I. A i)|M. f x) = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
using assms
apply induct
apply (auto intro!: set_integral_Un set_integrable_Un set_integrable_UN simp: disjoint_family_on_def)
by (subst set_integral_Un, auto intro: set_integrable_UN)
(* TODO: find a better name? *)
lemma pos_integrable_to_top:
fixes l::real
assumes "\<And>i. A i \<in> sets M" "mono A"
assumes nneg: "\<And>x i. x \<in> A i \<Longrightarrow> 0 \<le> f x"
and intgbl: "\<And>i::nat. set_integrable M (A i) f"
and lim: "(\<lambda>i::nat. LINT x:A i|M. f x) \<longlonglongrightarrow> l"
shows "set_integrable M (\<Union>i. A i) f"
apply (rule integrable_monotone_convergence[where f = "\<lambda>i::nat. \<lambda>x. indicator (A i) x *\<^sub>R f x" and x = l])
apply (rule intgbl)
prefer 3 apply (rule lim)
apply (rule AE_I2)
using \<open>mono A\<close> apply (auto simp: mono_def nneg split: split_indicator) []
proof (rule AE_I2)
{ fix x assume "x \<in> space M"
show "(\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x"
proof cases
assume "\<exists>i. x \<in> A i"
then guess i ..
then have *: "eventually (\<lambda>i. x \<in> A i) sequentially"
using \<open>x \<in> A i\<close> \<open>mono A\<close> by (auto simp: eventually_sequentially mono_def)
show ?thesis
apply (intro Lim_eventually)
using *
apply eventually_elim
apply (auto split: split_indicator)
done
qed auto }
then show "(\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R f x) \<in> borel_measurable M"
apply (rule borel_measurable_LIMSEQ_real)
apply assumption
apply (intro borel_measurable_integrable intgbl)
done
qed
(* Proof from Royden Real Analysis, p. 91. *)
lemma lebesgue_integral_countable_add:
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes meas[intro]: "\<And>i::nat. A i \<in> sets M"
and disj: "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
and intgbl: "set_integrable M (\<Union>i. A i) f"
shows "LINT x:(\<Union>i. A i)|M. f x = (\<Sum>i. (LINT x:(A i)|M. f x))"
proof (subst integral_suminf[symmetric])
show int_A: "\<And>i. set_integrable M (A i) f"
using intgbl by (rule set_integrable_subset) auto
{ fix x assume "x \<in> space M"
have "(\<lambda>i. indicator (A i) x *\<^sub>R f x) sums (indicator (\<Union>i. A i) x *\<^sub>R f x)"
by (intro sums_scaleR_left indicator_sums) fact }
note sums = this
have norm_f: "\<And>i. set_integrable M (A i) (\<lambda>x. norm (f x))"
using int_A[THEN integrable_norm] by auto
show "AE x in M. summable (\<lambda>i. norm (indicator (A i) x *\<^sub>R f x))"
using disj by (intro AE_I2) (auto intro!: summable_mult2 sums_summable[OF indicator_sums])
show "summable (\<lambda>i. LINT x|M. norm (indicator (A i) x *\<^sub>R f x))"
proof (rule summableI_nonneg_bounded)
fix n
show "0 \<le> LINT x|M. norm (indicator (A n) x *\<^sub>R f x)"
using norm_f by (auto intro!: integral_nonneg_AE)
have "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) =
(\<Sum>i<n. set_lebesgue_integral M (A i) (\<lambda>x. norm (f x)))"
by (simp add: abs_mult)
also have "\<dots> = set_lebesgue_integral M (\<Union>i<n. A i) (\<lambda>x. norm (f x))"
using norm_f
by (subst set_integral_finite_Union) (auto simp: disjoint_family_on_def disj)
also have "\<dots> \<le> set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
using intgbl[THEN integrable_norm]
by (intro integral_mono set_integrable_UN[of "{..<n}"] norm_f)
(auto split: split_indicator)
finally show "(\<Sum>i<n. LINT x|M. norm (indicator (A i) x *\<^sub>R f x)) \<le>
set_lebesgue_integral M (\<Union>i. A i) (\<lambda>x. norm (f x))"
by simp
qed
show "set_lebesgue_integral M (UNION UNIV A) f = LINT x|M. (\<Sum>i. indicator (A i) x *\<^sub>R f x)"
apply (rule Bochner_Integration.integral_cong[OF refl])
apply (subst suminf_scaleR_left[OF sums_summable[OF indicator_sums, OF disj], symmetric])
using sums_unique[OF indicator_sums[OF disj]]
apply auto
done
qed
lemma set_integral_cont_up:
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "incseq A"
and intgbl: "set_integrable M (\<Union>i. A i) f"
shows "(\<lambda>i. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Union>i. A i)|M. f x"
proof (intro integral_dominated_convergence[where w="\<lambda>x. indicator (\<Union>i. A i) x *\<^sub>R norm (f x)"])
have int_A: "\<And>i. set_integrable M (A i) f"
using intgbl by (rule set_integrable_subset) auto
then show "\<And>i. set_borel_measurable M (A i) f" "set_borel_measurable M (\<Union>i. A i) f"
"set_integrable M (\<Union>i. A i) (\<lambda>x. norm (f x))"
using intgbl integrable_norm[OF intgbl] by auto
{ fix x i assume "x \<in> A i"
with A have "(\<lambda>xa. indicator (A xa) x::real) \<longlonglongrightarrow> 1 \<longleftrightarrow> (\<lambda>xa. 1::real) \<longlonglongrightarrow> 1"
by (intro filterlim_cong refl)
(fastforce simp: eventually_sequentially incseq_def subset_eq intro!: exI[of _ i]) }
then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Union>i. A i) x *\<^sub>R f x"
by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
qed (auto split: split_indicator)
(* Can the int0 hypothesis be dropped? *)
lemma set_integral_cont_down:
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes [measurable]: "\<And>i. A i \<in> sets M" and A: "decseq A"
and int0: "set_integrable M (A 0) f"
shows "(\<lambda>i::nat. LINT x:(A i)|M. f x) \<longlonglongrightarrow> LINT x:(\<Inter>i. A i)|M. f x"
proof (rule integral_dominated_convergence)
have int_A: "\<And>i. set_integrable M (A i) f"
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
show "set_integrable M (A 0) (\<lambda>x. norm (f x))"
using int0[THEN integrable_norm] by simp
have "set_integrable M (\<Inter>i. A i) f"
using int0 by (rule set_integrable_subset) (insert A, auto simp: decseq_def)
with int_A show "set_borel_measurable M (\<Inter>i. A i) f" "\<And>i. set_borel_measurable M (A i) f"
by auto
show "\<And>i. AE x in M. norm (indicator (A i) x *\<^sub>R f x) \<le> indicator (A 0) x *\<^sub>R norm (f x)"
using A by (auto split: split_indicator simp: decseq_def)
{ fix x i assume "x \<in> space M" "x \<notin> A i"
with A have "(\<lambda>i. indicator (A i) x::real) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<lambda>i. 0::real) \<longlonglongrightarrow> 0"
by (intro filterlim_cong refl)
(auto split: split_indicator simp: eventually_sequentially decseq_def intro!: exI[of _ i]) }
then show "AE x in M. (\<lambda>i. indicator (A i) x *\<^sub>R f x) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x *\<^sub>R f x"
by (intro AE_I2 tendsto_intros) (auto split: split_indicator)
qed
lemma set_integral_at_point:
fixes a :: real
assumes "set_integrable M {a} f"
and [simp]: "{a} \<in> sets M" and "(emeasure M) {a} \<noteq> \<infinity>"
shows "(LINT x:{a} | M. f x) = f a * measure M {a}"
proof-
have "set_lebesgue_integral M {a} f = set_lebesgue_integral M {a} (%x. f a)"
by (intro set_lebesgue_integral_cong) simp_all
then show ?thesis using assms by simp
qed
abbreviation complex_integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
"complex_integrable M f \<equiv> integrable M f"
abbreviation complex_lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" ("integral\<^sup>C") where
"integral\<^sup>C M f == integral\<^sup>L M f"
syntax
"_complex_lebesgue_integral" :: "pttrn \<Rightarrow> complex \<Rightarrow> 'a measure \<Rightarrow> complex"
("\<integral>\<^sup>C _. _ \<partial>_" [60,61] 110)
translations
"\<integral>\<^sup>Cx. f \<partial>M" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
syntax
"_ascii_complex_lebesgue_integral" :: "pttrn \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
("(3CLINT _|_. _)" [0,110,60] 60)
translations
"CLINT x|M. f" == "CONST complex_lebesgue_integral M (\<lambda>x. f)"
lemma complex_integrable_cnj [simp]:
"complex_integrable M (\<lambda>x. cnj (f x)) \<longleftrightarrow> complex_integrable M f"
proof
assume "complex_integrable M (\<lambda>x. cnj (f x))"
then have "complex_integrable M (\<lambda>x. cnj (cnj (f x)))"
by (rule integrable_cnj)
then show "complex_integrable M f"
by simp
qed simp
lemma complex_of_real_integrable_eq:
"complex_integrable M (\<lambda>x. complex_of_real (f x)) \<longleftrightarrow> integrable M f"
proof
assume "complex_integrable M (\<lambda>x. complex_of_real (f x))"
then have "integrable M (\<lambda>x. Re (complex_of_real (f x)))"
by (rule integrable_Re)
then show "integrable M f"
by simp
qed simp
abbreviation complex_set_integrable :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> bool" where
"complex_set_integrable M A f \<equiv> set_integrable M A f"
abbreviation complex_set_lebesgue_integral :: "'a measure \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> complex) \<Rightarrow> complex" where
"complex_set_lebesgue_integral M A f \<equiv> set_lebesgue_integral M A f"
syntax
"_ascii_complex_set_lebesgue_integral" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'a measure \<Rightarrow> real \<Rightarrow> real"
("(4CLINT _:_|_. _)" [0,60,110,61] 60)
translations
"CLINT x:A|M. f" == "CONST complex_set_lebesgue_integral M A (\<lambda>x. f)"
lemma set_borel_measurable_continuous:
fixes f :: "_ \<Rightarrow> _::real_normed_vector"
assumes "S \<in> sets borel" "continuous_on S f"
shows "set_borel_measurable borel S f"
proof -
have "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable borel"
by (intro assms borel_measurable_continuous_on_if continuous_on_const)
also have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. indicator S x *\<^sub>R f x)"
by auto
finally show ?thesis .
qed
lemma set_measurable_continuous_on_ivl:
assumes "continuous_on {a..b} (f :: real \<Rightarrow> real)"
shows "set_borel_measurable borel {a..b} f"
by (rule set_borel_measurable_continuous[OF _ assms]) simp
end