(* Title: HOL/Probability/Lebesgue_Measure.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
Author: Jeremy Avigad
Author: Luke Serafin
*)
section {* Lebesgue measure *}
theory Lebesgue_Measure
imports Finite_Product_Measure Bochner_Integration Caratheodory
begin
subsection {* Every right continuous and nondecreasing function gives rise to a measure *}
definition interval_measure :: "(real \<Rightarrow> real) \<Rightarrow> real measure" where
"interval_measure F = extend_measure UNIV {(a, b). a \<le> b} (\<lambda>(a, b). {a <.. b}) (\<lambda>(a, b). ereal (F b - F a))"
lemma emeasure_interval_measure_Ioc:
assumes "a \<le> b"
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes right_cont_F : "\<And>a. continuous (at_right a) F"
shows "emeasure (interval_measure F) {a <.. b} = F b - F a"
proof (rule extend_measure_caratheodory_pair[OF interval_measure_def `a \<le> b`])
show "semiring_of_sets UNIV {{a<..b} |a b :: real. a \<le> b}"
proof (unfold_locales, safe)
fix a b c d :: real assume *: "a \<le> b" "c \<le> d"
then show "\<exists>C\<subseteq>{{a<..b} |a b. a \<le> b}. finite C \<and> disjoint C \<and> {a<..b} - {c<..d} = \<Union>C"
proof cases
let ?C = "{{a<..b}}"
assume "b < c \<or> d \<le> a \<or> d \<le> c"
with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
by (auto simp add: disjoint_def)
thus ?thesis ..
next
let ?C = "{{a<..c}, {d<..b}}"
assume "\<not> (b < c \<or> d \<le> a \<or> d \<le> c)"
with * have "?C \<subseteq> {{a<..b} |a b. a \<le> b} \<and> finite ?C \<and> disjoint ?C \<and> {a<..b} - {c<..d} = \<Union>?C"
by (auto simp add: disjoint_def Ioc_inj) (metis linear)+
thus ?thesis ..
qed
qed (auto simp: Ioc_inj, metis linear)
next
fix l r :: "nat \<Rightarrow> real" and a b :: real
assume l_r[simp]: "\<And>n. l n \<le> r n" and "a \<le> b" and disj: "disjoint_family (\<lambda>n. {l n<..r n})"
assume lr_eq_ab: "(\<Union>i. {l i<..r i}) = {a<..b}"
have [intro, simp]: "\<And>a b. a \<le> b \<Longrightarrow> 0 \<le> F b - F a"
by (auto intro!: l_r mono_F simp: diff_le_iff)
{ fix S :: "nat set" assume "finite S"
moreover note `a \<le> b`
moreover have "\<And>i. i \<in> S \<Longrightarrow> {l i <.. r i} \<subseteq> {a <.. b}"
unfolding lr_eq_ab[symmetric] by auto
ultimately have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<le> F b - F a"
proof (induction S arbitrary: a rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "\<exists>i\<in>S. l i < r i"
with `finite S` have "Min (l ` {i\<in>S. l i < r i}) \<in> l ` {i\<in>S. l i < r i}"
by (intro Min_in) auto
then obtain m where m: "m \<in> S" "l m < r m" "l m = Min (l ` {i\<in>S. l i < r i})"
by fastforce
have "(\<Sum>i\<in>S. F (r i) - F (l i)) = (F (r m) - F (l m)) + (\<Sum>i\<in>S - {m}. F (r i) - F (l i))"
using m psubset by (intro setsum.remove) auto
also have "(\<Sum>i\<in>S - {m}. F (r i) - F (l i)) \<le> F b - F (r m)"
proof (intro psubset.IH)
show "S - {m} \<subset> S"
using `m\<in>S` by auto
show "r m \<le> b"
using psubset.prems(2)[OF `m\<in>S`] `l m < r m` by auto
next
fix i assume "i \<in> S - {m}"
then have i: "i \<in> S" "i \<noteq> m" by auto
{ assume i': "l i < r i" "l i < r m"
moreover with `finite S` i m have "l m \<le> l i"
by auto
ultimately have "{l i <.. r i} \<inter> {l m <.. r m} \<noteq> {}"
by auto
then have False
using disjoint_family_onD[OF disj, of i m] i by auto }
then have "l i \<noteq> r i \<Longrightarrow> r m \<le> l i"
unfolding not_less[symmetric] using l_r[of i] by auto
then show "{l i <.. r i} \<subseteq> {r m <.. b}"
using psubset.prems(2)[OF `i\<in>S`] by auto
qed
also have "F (r m) - F (l m) \<le> F (r m) - F a"
using psubset.prems(2)[OF `m \<in> S`] `l m < r m`
by (auto simp add: Ioc_subset_iff intro!: mono_F)
finally show ?case
by (auto intro: add_mono)
qed (simp add: `a \<le> b` less_le)
qed }
note claim1 = this
(* second key induction: a lower bound on the measures of any finite collection of Ai's
that cover an interval {u..v} *)
{ fix S u v and l r :: "nat \<Rightarrow> real"
assume "finite S" "\<And>i. i\<in>S \<Longrightarrow> l i < r i" "{u..v} \<subseteq> (\<Union>i\<in>S. {l i<..< r i})"
then have "F v - F u \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
proof (induction arbitrary: v u rule: finite_psubset_induct)
case (psubset S)
show ?case
proof cases
assume "S = {}" then show ?case
using psubset by (simp add: mono_F)
next
assume "S \<noteq> {}"
then obtain j where "j \<in> S"
by auto
let ?R = "r j < u \<or> l j > v \<or> (\<exists>i\<in>S-{j}. l i \<le> l j \<and> r j \<le> r i)"
show ?case
proof cases
assume "?R"
with `j \<in> S` psubset.prems have "{u..v} \<subseteq> (\<Union>i\<in>S-{j}. {l i<..< r i})"
apply (auto simp: subset_eq Ball_def)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis Diff_iff less_le_trans leD linear singletonD)
apply (metis order_trans less_le_not_le linear)
done
with `j \<in> S` have "F v - F u \<le> (\<Sum>i\<in>S - {j}. F (r i) - F (l i))"
by (intro psubset) auto
also have "\<dots> \<le> (\<Sum>i\<in>S. F (r i) - F (l i))"
using psubset.prems
by (intro setsum_mono2 psubset) (auto intro: less_imp_le)
finally show ?thesis .
next
assume "\<not> ?R"
then have j: "u \<le> r j" "l j \<le> v" "\<And>i. i \<in> S - {j} \<Longrightarrow> r i < r j \<or> l i > l j"
by (auto simp: not_less)
let ?S1 = "{i \<in> S. l i < l j}"
let ?S2 = "{i \<in> S. r i > r j}"
have "(\<Sum>i\<in>S. F (r i) - F (l i)) \<ge> (\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i))"
using `j \<in> S` `finite S` psubset.prems j
by (intro setsum_mono2) (auto intro: less_imp_le)
also have "(\<Sum>i\<in>?S1 \<union> ?S2 \<union> {j}. F (r i) - F (l i)) =
(\<Sum>i\<in>?S1. F (r i) - F (l i)) + (\<Sum>i\<in>?S2 . F (r i) - F (l i)) + (F (r j) - F (l j))"
using psubset(1) psubset.prems(1) j
apply (subst setsum.union_disjoint)
apply simp_all
apply (subst setsum.union_disjoint)
apply auto
apply (metis less_le_not_le)
done
also (xtrans) have "(\<Sum>i\<in>?S1. F (r i) - F (l i)) \<ge> F (l j) - F u"
using `j \<in> S` `finite S` psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis less_le_trans not_le)
done
also (xtrans) have "(\<Sum>i\<in>?S2. F (r i) - F (l i)) \<ge> F v - F (r j)"
using `j \<in> S` `finite S` psubset.prems j
apply (intro psubset.IH psubset)
apply (auto simp: subset_eq Ball_def)
apply (metis le_less_trans not_le)
done
finally (xtrans) show ?case
by (auto simp: add_mono)
qed
qed
qed }
note claim2 = this
(* now prove the inequality going the other way *)
{ fix epsilon :: real assume egt0: "epsilon > 0"
have "\<forall>i. \<exists>d. d > 0 & F (r i + d) < F (r i) + epsilon / 2^(i+2)"
proof
fix i
note right_cont_F [of "r i"]
thus "\<exists>d. d > 0 \<and> F (r i + d) < F (r i) + epsilon / 2^(i+2)"
apply -
apply (subst (asm) continuous_at_right_real_increasing)
apply (rule mono_F, assumption)
apply (drule_tac x = "epsilon / 2 ^ (i + 2)" in spec)
apply (erule impE)
using egt0 by (auto simp add: field_simps)
qed
then obtain delta where
deltai_gt0: "\<And>i. delta i > 0" and
deltai_prop: "\<And>i. F (r i + delta i) < F (r i) + epsilon / 2^(i+2)"
by metis
have "\<exists>a' > a. F a' - F a < epsilon / 2"
apply (insert right_cont_F [of a])
apply (subst (asm) continuous_at_right_real_increasing)
using mono_F apply force
apply (drule_tac x = "epsilon / 2" in spec)
using egt0 apply (auto simp add: field_simps)
by (metis add_less_cancel_left comm_monoid_add_class.add.right_neutral
comm_semiring_1_class.normalizing_semiring_rules(24) mult_2 mult_2_right)
then obtain a' where a'lea [arith]: "a' > a" and
a_prop: "F a' - F a < epsilon / 2"
by auto
def S' \<equiv> "{i. l i < r i}"
obtain S :: "nat set" where
"S \<subseteq> S'" and finS: "finite S" and
Sprop: "{a'..b} \<subseteq> (\<Union>i \<in> S. {l i<..<r i + delta i})"
proof (rule compactE_image)
show "compact {a'..b}"
by (rule compact_Icc)
show "\<forall>i \<in> S'. open ({l i<..<r i + delta i})" by auto
have "{a'..b} \<subseteq> {a <.. b}"
by auto
also have "{a <.. b} = (\<Union>i\<in>S'. {l i<..r i})"
unfolding lr_eq_ab[symmetric] by (fastforce simp add: S'_def intro: less_le_trans)
also have "\<dots> \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})"
apply (intro UN_mono)
apply (auto simp: S'_def)
apply (cut_tac i=i in deltai_gt0)
apply simp
done
finally show "{a'..b} \<subseteq> (\<Union>i \<in> S'. {l i<..<r i + delta i})" .
qed
with S'_def have Sprop2: "\<And>i. i \<in> S \<Longrightarrow> l i < r i" by auto
from finS have "\<exists>n. \<forall>i \<in> S. i \<le> n"
by (subst finite_nat_set_iff_bounded_le [symmetric])
then obtain n where Sbound [rule_format]: "\<forall>i \<in> S. i \<le> n" ..
have "F b - F a' \<le> (\<Sum>i\<in>S. F (r i + delta i) - F (l i))"
apply (rule claim2 [rule_format])
using finS Sprop apply auto
apply (frule Sprop2)
apply (subgoal_tac "delta i > 0")
apply arith
by (rule deltai_gt0)
also have "... \<le> (SUM i : S. F(r i) - F(l i) + epsilon / 2^(i+2))"
apply (rule setsum_mono)
apply simp
apply (rule order_trans)
apply (rule less_imp_le)
apply (rule deltai_prop)
by auto
also have "... = (SUM i : S. F(r i) - F(l i)) +
(epsilon / 4) * (SUM i : S. (1 / 2)^i)" (is "_ = ?t + _")
by (subst setsum.distrib) (simp add: field_simps setsum_right_distrib)
also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
apply (rule add_left_mono)
apply (rule mult_left_mono)
apply (rule setsum_mono2)
using egt0 apply auto
by (frule Sbound, auto)
also have "... \<le> ?t + (epsilon / 2)"
apply (rule add_left_mono)
apply (subst geometric_sum)
apply auto
apply (rule mult_left_mono)
using egt0 apply auto
done
finally have aux2: "F b - F a' \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2"
by simp
have "F b - F a = (F b - F a') + (F a' - F a)"
by auto
also have "... \<le> (F b - F a') + epsilon / 2"
using a_prop by (intro add_left_mono) simp
also have "... \<le> (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon / 2 + epsilon / 2"
apply (intro add_right_mono)
apply (rule aux2)
done
also have "... = (\<Sum>i\<in>S. F (r i) - F (l i)) + epsilon"
by auto
also have "... \<le> (\<Sum>i\<le>n. F (r i) - F (l i)) + epsilon"
using finS Sbound Sprop by (auto intro!: add_right_mono setsum_mono3)
finally have "ereal (F b - F a) \<le> (\<Sum>i\<le>n. ereal (F (r i) - F (l i))) + epsilon"
by simp
then have "ereal (F b - F a) \<le> (\<Sum>i. ereal (F (r i) - F (l i))) + (epsilon :: real)"
apply (rule_tac order_trans)
prefer 2
apply (rule add_mono[where c="ereal epsilon"])
apply (rule suminf_upper[of _ "Suc n"])
apply (auto simp add: lessThan_Suc_atMost)
done }
hence "ereal (F b - F a) \<le> (\<Sum>i. ereal (F (r i) - F (l i)))"
by (auto intro: ereal_le_epsilon2)
moreover
have "(\<Sum>i. ereal (F (r i) - F (l i))) \<le> ereal (F b - F a)"
by (auto simp add: claim1 intro!: suminf_bound)
ultimately show "(\<Sum>n. ereal (F (r n) - F (l n))) = ereal (F b - F a)"
by simp
qed (auto simp: Ioc_inj diff_le_iff mono_F)
lemma measure_interval_measure_Ioc:
assumes "a \<le> b"
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes right_cont_F : "\<And>a. continuous (at_right a) F"
shows "measure (interval_measure F) {a <.. b} = F b - F a"
unfolding measure_def
apply (subst emeasure_interval_measure_Ioc)
apply fact+
apply simp
done
lemma emeasure_interval_measure_Ioc_eq:
"(\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y) \<Longrightarrow> (\<And>a. continuous (at_right a) F) \<Longrightarrow>
emeasure (interval_measure F) {a <.. b} = (if a \<le> b then F b - F a else 0)"
using emeasure_interval_measure_Ioc[of a b F] by auto
lemma sets_interval_measure [simp, measurable_cong]: "sets (interval_measure F) = sets borel"
apply (simp add: sets_extend_measure interval_measure_def borel_sigma_sets_Ioc)
apply (rule sigma_sets_eqI)
apply auto
apply (case_tac "a \<le> ba")
apply (auto intro: sigma_sets.Empty)
done
lemma space_interval_measure [simp]: "space (interval_measure F) = UNIV"
by (simp add: interval_measure_def space_extend_measure)
lemma emeasure_interval_measure_Icc:
assumes "a \<le> b"
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes cont_F : "continuous_on UNIV F"
shows "emeasure (interval_measure F) {a .. b} = F b - F a"
proof (rule tendsto_unique)
{ fix a b :: real assume "a \<le> b" then have "emeasure (interval_measure F) {a <.. b} = F b - F a"
using cont_F
by (subst emeasure_interval_measure_Ioc)
(auto intro: mono_F continuous_within_subset simp: continuous_on_eq_continuous_within) }
note * = this
let ?F = "interval_measure F"
show "((\<lambda>a. F b - F a) ---> emeasure ?F {a..b}) (at_left a)"
proof (rule tendsto_at_left_sequentially)
show "a - 1 < a" by simp
fix X assume "\<And>n. X n < a" "incseq X" "X ----> a"
with `a \<le> b` have "(\<lambda>n. emeasure ?F {X n<..b}) ----> emeasure ?F (\<Inter>n. {X n <..b})"
apply (intro Lim_emeasure_decseq)
apply (auto simp: decseq_def incseq_def emeasure_interval_measure_Ioc *)
apply force
apply (subst (asm ) *)
apply (auto intro: less_le_trans less_imp_le)
done
also have "(\<Inter>n. {X n <..b}) = {a..b}"
using `\<And>n. X n < a`
apply auto
apply (rule LIMSEQ_le_const2[OF `X ----> a`])
apply (auto intro: less_imp_le)
apply (auto intro: less_le_trans)
done
also have "(\<lambda>n. emeasure ?F {X n<..b}) = (\<lambda>n. F b - F (X n))"
using `\<And>n. X n < a` `a \<le> b` by (subst *) (auto intro: less_imp_le less_le_trans)
finally show "(\<lambda>n. F b - F (X n)) ----> emeasure ?F {a..b}" .
qed
show "((\<lambda>a. ereal (F b - F a)) ---> F b - F a) (at_left a)"
using cont_F
by (intro lim_ereal[THEN iffD2] tendsto_intros )
(auto simp: continuous_on_def intro: tendsto_within_subset)
qed (rule trivial_limit_at_left_real)
lemma sigma_finite_interval_measure:
assumes mono_F: "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"
assumes right_cont_F : "\<And>a. continuous (at_right a) F"
shows "sigma_finite_measure (interval_measure F)"
apply unfold_locales
apply (intro exI[of _ "(\<lambda>(a, b). {a <.. b}) ` (\<rat> \<times> \<rat>)"])
apply (auto intro!: Rats_no_top_le Rats_no_bot_less countable_rat simp: emeasure_interval_measure_Ioc_eq[OF assms])
done
subsection {* Lebesgue-Borel measure *}
definition lborel :: "('a :: euclidean_space) measure" where
"lborel = distr (\<Pi>\<^sub>M b\<in>Basis. interval_measure (\<lambda>x. x)) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
lemma
shows sets_lborel[simp, measurable_cong]: "sets lborel = sets borel"
and space_lborel[simp]: "space lborel = space borel"
and measurable_lborel1[simp]: "measurable M lborel = measurable M borel"
and measurable_lborel2[simp]: "measurable lborel M = measurable borel M"
by (simp_all add: lborel_def)
context
begin
interpretation sigma_finite_measure "interval_measure (\<lambda>x. x)"
by (rule sigma_finite_interval_measure) auto
interpretation finite_product_sigma_finite "\<lambda>_. interval_measure (\<lambda>x. x)" Basis
proof qed simp
lemma lborel_eq_real: "lborel = interval_measure (\<lambda>x. x)"
unfolding lborel_def Basis_real_def
using distr_id[of "interval_measure (\<lambda>x. x)"]
by (subst distr_component[symmetric])
(simp_all add: distr_distr comp_def del: distr_id cong: distr_cong)
lemma lborel_eq: "lborel = distr (\<Pi>\<^sub>M b\<in>Basis. lborel) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
by (subst lborel_def) (simp add: lborel_eq_real)
lemma nn_integral_lborel_setprod:
assumes [measurable]: "\<And>b. b \<in> Basis \<Longrightarrow> f b \<in> borel_measurable borel"
assumes nn[simp]: "\<And>b x. b \<in> Basis \<Longrightarrow> 0 \<le> f b x"
shows "(\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. f b (x \<bullet> b)) \<partial>lborel) = (\<Prod>b\<in>Basis. (\<integral>\<^sup>+x. f b x \<partial>lborel))"
by (simp add: lborel_def nn_integral_distr product_nn_integral_setprod
product_nn_integral_singleton)
lemma emeasure_lborel_Icc[simp]:
fixes l u :: real
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l .. u} = u - l"
proof -
have "((\<lambda>f. f 1) -` {l..u} \<inter> space (Pi\<^sub>M {1} (\<lambda>b. interval_measure (\<lambda>x. x)))) = {1::real} \<rightarrow>\<^sub>E {l..u}"
by (auto simp: space_PiM)
then show ?thesis
by (simp add: lborel_def emeasure_distr emeasure_PiM emeasure_interval_measure_Icc continuous_on_id)
qed
lemma emeasure_lborel_Icc_eq: "emeasure lborel {l .. u} = ereal (if l \<le> u then u - l else 0)"
by simp
lemma emeasure_lborel_cbox[simp]:
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows "emeasure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b) :: ereal) = indicator (cbox l u)"
by (auto simp: fun_eq_iff cbox_def setprod_ereal_0 split: split_indicator)
then have "emeasure lborel (cbox l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b .. u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
by simp
also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (subst nn_integral_lborel_setprod) (simp_all add: setprod_ereal inner_diff_left)
finally show ?thesis .
qed
lemma AE_lborel_singleton: "AE x in lborel::'a::euclidean_space measure. x \<noteq> c"
using SOME_Basis AE_discrete_difference [of "{c}" lborel]
emeasure_lborel_cbox [of c c] by (auto simp add: cbox_sing)
lemma emeasure_lborel_Ioo[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l <..< u} = ereal (u - l)"
proof -
have "emeasure lborel {l <..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed
lemma emeasure_lborel_Ioc[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l <.. u} = ereal (u - l)"
proof -
have "emeasure lborel {l <.. u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed
lemma emeasure_lborel_Ico[simp]:
assumes [simp]: "l \<le> u"
shows "emeasure lborel {l ..< u} = ereal (u - l)"
proof -
have "emeasure lborel {l ..< u} = emeasure lborel {l .. u}"
using AE_lborel_singleton[of u] AE_lborel_singleton[of l] by (intro emeasure_eq_AE) auto
then show ?thesis
by simp
qed
lemma emeasure_lborel_box[simp]:
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows "emeasure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
proof -
have "(\<lambda>x. \<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b) :: ereal) = indicator (box l u)"
by (auto simp: fun_eq_iff box_def setprod_ereal_0 split: split_indicator)
then have "emeasure lborel (box l u) = (\<integral>\<^sup>+x. (\<Prod>b\<in>Basis. indicator {l\<bullet>b <..< u\<bullet>b} (x \<bullet> b)) \<partial>lborel)"
by simp
also have "\<dots> = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (subst nn_integral_lborel_setprod) (simp_all add: setprod_ereal inner_diff_left)
finally show ?thesis .
qed
lemma emeasure_lborel_cbox_eq:
"emeasure lborel (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(2)[THEN iffD2, of u l] by (auto simp: not_le)
lemma emeasure_lborel_box_eq:
"emeasure lborel (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then \<Prod>b\<in>Basis. (u - l) \<bullet> b else 0)"
using box_eq_empty(1)[THEN iffD2, of u l] by (auto simp: not_le dest!: less_imp_le) force
lemma
fixes l u :: real
assumes [simp]: "l \<le> u"
shows measure_lborel_Icc[simp]: "measure lborel {l .. u} = u - l"
and measure_lborel_Ico[simp]: "measure lborel {l ..< u} = u - l"
and measure_lborel_Ioc[simp]: "measure lborel {l <.. u} = u - l"
and measure_lborel_Ioo[simp]: "measure lborel {l <..< u} = u - l"
by (simp_all add: measure_def)
lemma
assumes [simp]: "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
shows measure_lborel_box[simp]: "measure lborel (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
and measure_lborel_cbox[simp]: "measure lborel (cbox l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
by (simp_all add: measure_def)
lemma sigma_finite_lborel: "sigma_finite_measure lborel"
proof
show "\<exists>A::'a set set. countable A \<and> A \<subseteq> sets lborel \<and> \<Union>A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
by (intro exI[of _ "range (\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One))"])
(auto simp: emeasure_lborel_cbox_eq UN_box_eq_UNIV)
qed
end
lemma emeasure_lborel_UNIV: "emeasure lborel (UNIV::'a::euclidean_space set) = \<infinity>"
unfolding UN_box_eq_UNIV[symmetric]
apply (subst SUP_emeasure_incseq[symmetric])
apply (auto simp: incseq_def subset_box inner_add_left setprod_constant intro!: SUP_PInfty)
apply (rule_tac x="Suc n" in exI)
apply (rule order_trans[OF _ self_le_power])
apply (auto simp: card_gt_0_iff real_of_nat_Suc)
done
lemma emeasure_lborel_singleton[simp]: "emeasure lborel {x} = 0"
using emeasure_lborel_cbox[of x x] nonempty_Basis
by (auto simp del: emeasure_lborel_cbox nonempty_Basis simp add: cbox_sing)
lemma emeasure_lborel_countable:
fixes A :: "'a::euclidean_space set"
assumes "countable A"
shows "emeasure lborel A = 0"
proof -
have "A \<subseteq> (\<Union>i. {from_nat_into A i})" using from_nat_into_surj assms by force
moreover have "emeasure lborel (\<Union>i. {from_nat_into A i}) = 0"
by (rule emeasure_UN_eq_0) auto
ultimately have "emeasure lborel A \<le> 0" using emeasure_mono
by (metis assms bot.extremum_unique emeasure_empty image_eq_UN range_from_nat_into sets.empty_sets)
thus ?thesis by (auto simp add: emeasure_le_0_iff)
qed
lemma countable_imp_null_set_lborel: "countable A \<Longrightarrow> A \<in> null_sets lborel"
by (simp add: null_sets_def emeasure_lborel_countable sets.countable)
lemma finite_imp_null_set_lborel: "finite A \<Longrightarrow> A \<in> null_sets lborel"
by (intro countable_imp_null_set_lborel countable_finite)
lemma lborel_neq_count_space[simp]: "lborel \<noteq> count_space (A::('a::ordered_euclidean_space) set)"
proof
assume asm: "lborel = count_space A"
have "space lborel = UNIV" by simp
hence [simp]: "A = UNIV" by (subst (asm) asm) (simp only: space_count_space)
have "emeasure lborel {undefined::'a} = 1"
by (subst asm, subst emeasure_count_space_finite) auto
moreover have "emeasure lborel {undefined} \<noteq> 1" by simp
ultimately show False by contradiction
qed
subsection {* Affine transformation on the Lebesgue-Borel *}
lemma lborel_eqI:
fixes M :: "'a::euclidean_space measure"
assumes emeasure_eq: "\<And>l u. (\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b) \<Longrightarrow> emeasure M (box l u) = (\<Prod>b\<in>Basis. (u - l) \<bullet> b)"
assumes sets_eq: "sets M = sets borel"
shows "lborel = M"
proof (rule measure_eqI_generator_eq)
let ?E = "range (\<lambda>(a, b). box a b::'a set)"
show "Int_stable ?E"
by (auto simp: Int_stable_def box_Int_box)
show "?E \<subseteq> Pow UNIV" "sets lborel = sigma_sets UNIV ?E" "sets M = sigma_sets UNIV ?E"
by (simp_all add: borel_eq_box sets_eq)
let ?A = "\<lambda>n::nat. box (- (real n *\<^sub>R One)) (real n *\<^sub>R One) :: 'a set"
show "range ?A \<subseteq> ?E" "(\<Union>i. ?A i) = UNIV"
unfolding UN_box_eq_UNIV by auto
{ fix i show "emeasure lborel (?A i) \<noteq> \<infinity>" by auto }
{ fix X assume "X \<in> ?E" then show "emeasure lborel X = emeasure M X"
apply (auto simp: emeasure_eq emeasure_lborel_box_eq )
apply (subst box_eq_empty(1)[THEN iffD2])
apply (auto intro: less_imp_le simp: not_le)
done }
qed
lemma lborel_affine:
fixes t :: "'a::euclidean_space" assumes "c \<noteq> 0"
shows "lborel = density (distr lborel borel (\<lambda>x. t + c *\<^sub>R x)) (\<lambda>_. \<bar>c\<bar>^DIM('a))" (is "_ = ?D")
proof (rule lborel_eqI)
let ?B = "Basis :: 'a set"
fix l u assume le: "\<And>b. b \<in> ?B \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b"
show "emeasure ?D (box l u) = (\<Prod>b\<in>?B. (u - l) \<bullet> b)"
proof cases
assume "0 < c"
then have "(\<lambda>x. t + c *\<^sub>R x) -` box l u = box ((l - t) /\<^sub>R c) ((u - t) /\<^sub>R c)"
by (auto simp: field_simps box_def inner_simps)
with `0 < c` show ?thesis
using le
by (auto simp: field_simps emeasure_density nn_integral_distr nn_integral_cmult
emeasure_lborel_box_eq inner_simps setprod_dividef setprod_constant
borel_measurable_indicator' emeasure_distr)
next
assume "\<not> 0 < c" with `c \<noteq> 0` have "c < 0" by auto
then have "box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c) = (\<lambda>x. t + c *\<^sub>R x) -` box l u"
by (auto simp: field_simps box_def inner_simps)
then have "\<And>x. indicator (box l u) (t + c *\<^sub>R x) = (indicator (box ((u - t) /\<^sub>R c) ((l - t) /\<^sub>R c)) x :: ereal)"
by (auto split: split_indicator)
moreover
{ have "(- c) ^ card ?B * (\<Prod>x\<in>?B. l \<bullet> x - u \<bullet> x) =
(-1 * c) ^ card ?B * (\<Prod>x\<in>?B. -1 * (u \<bullet> x - l \<bullet> x))"
by simp
also have "\<dots> = (-1 * -1)^card ?B * c ^ card ?B * (\<Prod>x\<in>?B. u \<bullet> x - l \<bullet> x)"
unfolding setprod.distrib power_mult_distrib by (simp add: setprod_constant)
finally have "(- c) ^ card ?B * (\<Prod>x\<in>?B. l \<bullet> x - u \<bullet> x) = c ^ card ?B * (\<Prod>b\<in>?B. u \<bullet> b - l \<bullet> b)"
by simp }
ultimately show ?thesis
using `c < 0` le
by (auto simp: field_simps emeasure_density nn_integral_distr nn_integral_cmult
emeasure_lborel_box_eq inner_simps setprod_dividef setprod_constant
borel_measurable_indicator' emeasure_distr)
qed
qed simp
lemma lborel_real_affine:
"c \<noteq> 0 \<Longrightarrow> lborel = density (distr lborel borel (\<lambda>x. t + c * x)) (\<lambda>_. \<bar>c\<bar>)"
using lborel_affine[of c t] by simp
lemma AE_borel_affine:
fixes P :: "real \<Rightarrow> bool"
shows "c \<noteq> 0 \<Longrightarrow> Measurable.pred borel P \<Longrightarrow> AE x in lborel. P x \<Longrightarrow> AE x in lborel. P (t + c * x)"
by (subst lborel_real_affine[where t="- t / c" and c="1 / c"])
(simp_all add: AE_density AE_distr_iff field_simps)
lemma nn_integral_real_affine:
fixes c :: real assumes [measurable]: "f \<in> borel_measurable borel" and c: "c \<noteq> 0"
shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = \<bar>c\<bar> * (\<integral>\<^sup>+x. f (t + c * x) \<partial>lborel)"
by (subst lborel_real_affine[OF c, of t])
(simp add: nn_integral_density nn_integral_distr nn_integral_cmult)
lemma lborel_integrable_real_affine:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes f: "integrable lborel f"
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x))"
using f f[THEN borel_measurable_integrable] unfolding integrable_iff_bounded
by (subst (asm) nn_integral_real_affine[where c=c and t=t]) auto
lemma lborel_integrable_real_affine_iff:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
shows "c \<noteq> 0 \<Longrightarrow> integrable lborel (\<lambda>x. f (t + c * x)) \<longleftrightarrow> integrable lborel f"
using
lborel_integrable_real_affine[of f c t]
lborel_integrable_real_affine[of "\<lambda>x. f (t + c * x)" "1/c" "-t/c"]
by (auto simp add: field_simps)
lemma lborel_integral_real_affine:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}" and c :: real
assumes c: "c \<noteq> 0" shows "(\<integral>x. f x \<partial> lborel) = \<bar>c\<bar> *\<^sub>R (\<integral>x. f (t + c * x) \<partial>lborel)"
proof cases
assume f[measurable]: "integrable lborel f" then show ?thesis
using c f f[THEN borel_measurable_integrable] f[THEN lborel_integrable_real_affine, of c t]
by (subst lborel_real_affine[OF c, of t])
(simp add: integral_density integral_distr)
next
assume "\<not> integrable lborel f" with c show ?thesis
by (simp add: lborel_integrable_real_affine_iff not_integrable_integral_eq)
qed
lemma divideR_right:
fixes x y :: "'a::real_normed_vector"
shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
using scaleR_cancel_left[of r y "x /\<^sub>R r"] by simp
lemma lborel_has_bochner_integral_real_affine_iff:
fixes x :: "'a :: {banach, second_countable_topology}"
shows "c \<noteq> 0 \<Longrightarrow>
has_bochner_integral lborel f x \<longleftrightarrow>
has_bochner_integral lborel (\<lambda>x. f (t + c * x)) (x /\<^sub>R \<bar>c\<bar>)"
unfolding has_bochner_integral_iff lborel_integrable_real_affine_iff
by (simp_all add: lborel_integral_real_affine[symmetric] divideR_right cong: conj_cong)
lemma lborel_distr_uminus: "distr lborel borel uminus = (lborel :: real measure)"
by (subst lborel_real_affine[of "-1" 0])
(auto simp: density_1 one_ereal_def[symmetric])
lemma lborel_distr_mult:
assumes "(c::real) \<noteq> 0"
shows "distr lborel borel (op * c) = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
proof-
have "distr lborel borel (op * c) = distr lborel lborel (op * c)" by (simp cong: distr_cong)
also from assms have "... = density lborel (\<lambda>_. inverse \<bar>c\<bar>)"
by (subst lborel_real_affine[of "inverse c" 0]) (auto simp: o_def distr_density_distr)
finally show ?thesis .
qed
lemma lborel_distr_mult':
assumes "(c::real) \<noteq> 0"
shows "lborel = density (distr lborel borel (op * c)) (\<lambda>_. abs c)"
proof-
have "lborel = density lborel (\<lambda>_. 1)" by (rule density_1[symmetric])
also from assms have "(\<lambda>_. 1 :: ereal) = (\<lambda>_. inverse (abs c) * abs c)" by (intro ext) simp
also have "density lborel ... = density (density lborel (\<lambda>_. inverse (abs c))) (\<lambda>_. abs c)"
by (subst density_density_eq) auto
also from assms have "density lborel (\<lambda>_. inverse (abs c)) = distr lborel borel (op * c)"
by (rule lborel_distr_mult[symmetric])
finally show ?thesis .
qed
lemma lborel_distr_plus: "distr lborel borel (op + c) = (lborel :: real measure)"
by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ereal_def[symmetric])
interpretation lborel!: sigma_finite_measure lborel
by (rule sigma_finite_lborel)
interpretation lborel_pair: pair_sigma_finite lborel lborel ..
lemma lborel_prod:
"lborel \<Otimes>\<^sub>M lborel = (lborel :: ('a::euclidean_space \<times> 'b::euclidean_space) measure)"
proof (rule lborel_eqI[symmetric], clarify)
fix la ua :: 'a and lb ub :: 'b
assume lu: "\<And>a b. (a, b) \<in> Basis \<Longrightarrow> (la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
have [simp]:
"\<And>b. b \<in> Basis \<Longrightarrow> la \<bullet> b \<le> ua \<bullet> b"
"\<And>b. b \<in> Basis \<Longrightarrow> lb \<bullet> b \<le> ub \<bullet> b"
"inj_on (\<lambda>u. (u, 0)) Basis" "inj_on (\<lambda>u. (0, u)) Basis"
"(\<lambda>u. (u, 0)) ` Basis \<inter> (\<lambda>u. (0, u)) ` Basis = {}"
"box (la, lb) (ua, ub) = box la ua \<times> box lb ub"
using lu[of _ 0] lu[of 0] by (auto intro!: inj_onI simp add: Basis_prod_def ball_Un box_def)
show "emeasure (lborel \<Otimes>\<^sub>M lborel) (box (la, lb) (ua, ub)) =
ereal (setprod (op \<bullet> ((ua, ub) - (la, lb))) Basis)"
by (simp add: lborel.emeasure_pair_measure_Times Basis_prod_def setprod.union_disjoint
setprod.reindex)
qed (simp add: borel_prod[symmetric])
(* FIXME: conversion in measurable prover *)
lemma lborelD_Collect[measurable (raw)]: "{x\<in>space borel. P x} \<in> sets borel \<Longrightarrow> {x\<in>space lborel. P x} \<in> sets lborel" by simp
lemma lborelD[measurable (raw)]: "A \<in> sets borel \<Longrightarrow> A \<in> sets lborel" by simp
subsection {* Equivalence Lebesgue integral on @{const lborel} and HK-integral *}
lemma has_integral_measure_lborel:
fixes A :: "'a::euclidean_space set"
assumes A[measurable]: "A \<in> sets borel" and finite: "emeasure lborel A < \<infinity>"
shows "((\<lambda>x. 1) has_integral measure lborel A) A"
proof -
{ fix l u :: 'a
have "((\<lambda>x. 1) has_integral measure lborel (box l u)) (box l u)"
proof cases
assume "\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b"
then show ?thesis
apply simp
apply (subst has_integral_restrict[symmetric, OF box_subset_cbox])
apply (subst has_integral_spike_interior_eq[where g="\<lambda>_. 1"])
using has_integral_const[of "1::real" l u]
apply (simp_all add: inner_diff_left[symmetric] content_cbox_cases)
done
next
assume "\<not> (\<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b)"
then have "box l u = {}"
unfolding box_eq_empty by (auto simp: not_le intro: less_imp_le)
then show ?thesis
by simp
qed }
note has_integral_box = this
{ fix a b :: 'a let ?M = "\<lambda>A. measure lborel (A \<inter> box a b)"
have "Int_stable (range (\<lambda>(a, b). box a b))"
by (auto simp: Int_stable_def box_Int_box)
moreover have "(range (\<lambda>(a, b). box a b)) \<subseteq> Pow UNIV"
by auto
moreover have "A \<in> sigma_sets UNIV (range (\<lambda>(a, b). box a b))"
using A unfolding borel_eq_box by simp
ultimately have "((\<lambda>x. 1) has_integral ?M A) (A \<inter> box a b)"
proof (induction rule: sigma_sets_induct_disjoint)
case (basic A) then show ?case
by (auto simp: box_Int_box has_integral_box)
next
case empty then show ?case
by simp
next
case (compl A)
then have [measurable]: "A \<in> sets borel"
by (simp add: borel_eq_box)
have "((\<lambda>x. 1) has_integral ?M (box a b)) (box a b)"
by (simp add: has_integral_box)
moreover have "((\<lambda>x. if x \<in> A \<inter> box a b then 1 else 0) has_integral ?M A) (box a b)"
by (subst has_integral_restrict) (auto intro: compl)
ultimately have "((\<lambda>x. 1 - (if x \<in> A \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
by (rule has_integral_sub)
then have "((\<lambda>x. (if x \<in> (UNIV - A) \<inter> box a b then 1 else 0)) has_integral ?M (box a b) - ?M A) (box a b)"
by (rule has_integral_eq_eq[THEN iffD1, rotated 1]) auto
then have "((\<lambda>x. 1) has_integral ?M (box a b) - ?M A) ((UNIV - A) \<inter> box a b)"
by (subst (asm) has_integral_restrict) auto
also have "?M (box a b) - ?M A = ?M (UNIV - A)"
by (subst measure_Diff[symmetric]) (auto simp: emeasure_lborel_box_eq Diff_Int_distrib2)
finally show ?case .
next
case (union F)
then have [measurable]: "\<And>i. F i \<in> sets borel"
by (simp add: borel_eq_box subset_eq)
have "((\<lambda>x. if x \<in> UNION UNIV F \<inter> box a b then 1 else 0) has_integral ?M (\<Union>i. F i)) (box a b)"
proof (rule has_integral_monotone_convergence_increasing)
let ?f = "\<lambda>k x. \<Sum>i<k. if x \<in> F i \<inter> box a b then 1 else 0 :: real"
show "\<And>k. (?f k has_integral (\<Sum>i<k. ?M (F i))) (box a b)"
using union.IH by (auto intro!: has_integral_setsum simp del: Int_iff)
show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
by (intro setsum_mono2) auto
from union(1) have *: "\<And>x i j. x \<in> F i \<Longrightarrow> x \<in> F j \<longleftrightarrow> j = i"
by (auto simp add: disjoint_family_on_def)
show "\<And>x. (\<lambda>k. ?f k x) ----> (if x \<in> UNION UNIV F \<inter> box a b then 1 else 0)"
apply (auto simp: * setsum.If_cases Iio_Int_singleton)
apply (rule_tac k="Suc xa" in LIMSEQ_offset)
apply (simp add: tendsto_const)
done
have *: "emeasure lborel ((\<Union>x. F x) \<inter> box a b) \<le> emeasure lborel (box a b)"
by (intro emeasure_mono) auto
with union(1) show "(\<lambda>k. \<Sum>i<k. ?M (F i)) ----> ?M (\<Union>i. F i)"
unfolding sums_def[symmetric] UN_extend_simps
by (intro measure_UNION) (auto simp: disjoint_family_on_def emeasure_lborel_box_eq)
qed
then show ?case
by (subst (asm) has_integral_restrict) auto
qed }
note * = this
show ?thesis
proof (rule has_integral_monotone_convergence_increasing)
let ?B = "\<lambda>n::nat. box (- real n *\<^sub>R One) (real n *\<^sub>R One) :: 'a set"
let ?f = "\<lambda>n::nat. \<lambda>x. if x \<in> A \<inter> ?B n then 1 else 0 :: real"
let ?M = "\<lambda>n. measure lborel (A \<inter> ?B n)"
show "\<And>n::nat. (?f n has_integral ?M n) A"
using * by (subst has_integral_restrict) simp_all
show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
by (auto simp: box_def)
{ fix x assume "x \<in> A"
moreover have "(\<lambda>k. indicator (A \<inter> ?B k) x :: real) ----> indicator (\<Union>k::nat. A \<inter> ?B k) x"
by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
ultimately show "(\<lambda>k. if x \<in> A \<inter> ?B k then 1 else 0::real) ----> 1"
by (simp add: indicator_def UN_box_eq_UNIV) }
have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) ----> emeasure lborel (\<Union>n::nat. A \<inter> ?B n)"
by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
also have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) = (\<lambda>n. measure lborel (A \<inter> ?B n))"
proof (intro ext emeasure_eq_ereal_measure)
fix n have "emeasure lborel (A \<inter> ?B n) \<le> emeasure lborel (?B n)"
by (intro emeasure_mono) auto
then show "emeasure lborel (A \<inter> ?B n) \<noteq> \<infinity>"
by auto
qed
finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) ----> measure lborel A"
using emeasure_eq_ereal_measure[of lborel A] finite
by (simp add: UN_box_eq_UNIV)
qed
qed
lemma nn_integral_has_integral:
fixes f::"'a::euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) = ereal r"
shows "(f has_integral r) UNIV"
using f proof (induct arbitrary: r rule: borel_measurable_induct_real)
case (set A)
moreover then have "((\<lambda>x. 1) has_integral measure lborel A) A"
by (intro has_integral_measure_lborel) (auto simp: ereal_indicator)
ultimately show ?case
by (simp add: ereal_indicator measure_def) (simp add: indicator_def)
next
case (mult g c)
then have "ereal c * (\<integral>\<^sup>+ x. g x \<partial>lborel) = ereal r"
by (subst nn_integral_cmult[symmetric]) auto
then obtain r' where "(c = 0 \<and> r = 0) \<or> ((\<integral>\<^sup>+ x. ereal (g x) \<partial>lborel) = ereal r' \<and> r = c * r')"
by (cases "\<integral>\<^sup>+ x. ereal (g x) \<partial>lborel") (auto split: split_if_asm)
with mult show ?case
by (auto intro!: has_integral_cmult_real)
next
case (add g h)
moreover
then have "(\<integral>\<^sup>+ x. h x + g x \<partial>lborel) = (\<integral>\<^sup>+ x. h x \<partial>lborel) + (\<integral>\<^sup>+ x. g x \<partial>lborel)"
unfolding plus_ereal.simps[symmetric] by (subst nn_integral_add) auto
with add obtain a b where "(\<integral>\<^sup>+ x. h x \<partial>lborel) = ereal a" "(\<integral>\<^sup>+ x. g x \<partial>lborel) = ereal b" "r = a + b"
by (cases "\<integral>\<^sup>+ x. h x \<partial>lborel" "\<integral>\<^sup>+ x. g x \<partial>lborel" rule: ereal2_cases) auto
ultimately show ?case
by (auto intro!: has_integral_add)
next
case (seq U)
note seq(1)[measurable] and f[measurable]
{ fix i x
have "U i x \<le> f x"
using seq(5)
apply (rule LIMSEQ_le_const)
using seq(4)
apply (auto intro!: exI[of _ i] simp: incseq_def le_fun_def)
done }
note U_le_f = this
{ fix i
have "(\<integral>\<^sup>+x. ereal (U i x) \<partial>lborel) \<le> (\<integral>\<^sup>+x. ereal (f x) \<partial>lborel)"
using U_le_f by (intro nn_integral_mono) simp
then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lborel) = ereal p" "p \<le> r"
using seq(6) by (cases "\<integral>\<^sup>+x. U i x \<partial>lborel") auto
moreover then have "0 \<le> p"
by (metis ereal_less_eq(5) nn_integral_nonneg)
moreover note seq
ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lborel) = ereal p \<and> 0 \<le> p \<and> p \<le> r \<and> (U i has_integral p) UNIV"
by auto }
then obtain p where p: "\<And>i. (\<integral>\<^sup>+x. ereal (U i x) \<partial>lborel) = ereal (p i)"
and bnd: "\<And>i. p i \<le> r" "\<And>i. 0 \<le> p i"
and U_int: "\<And>i.(U i has_integral (p i)) UNIV" by metis
have int_eq: "\<And>i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
have *: "f integrable_on UNIV \<and> (\<lambda>k. integral UNIV (U k)) ----> integral UNIV f"
proof (rule monotone_convergence_increasing)
show "\<forall>k. U k integrable_on UNIV" using U_int by auto
show "\<forall>k. \<forall>x\<in>UNIV. U k x \<le> U (Suc k) x" using `incseq U` by (auto simp: incseq_def le_fun_def)
then show "bounded {integral UNIV (U k) |k. True}"
using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
show "\<forall>x\<in>UNIV. (\<lambda>k. U k x) ----> f x"
using seq by auto
qed
moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) ----> (\<integral>\<^sup>+x. f x \<partial>lborel)"
using seq U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
ultimately have "integral UNIV f = r"
by (auto simp add: int_eq p seq intro: LIMSEQ_unique)
with * show ?case
by (simp add: has_integral_integral)
qed
lemma nn_integral_lborel_eq_integral:
fixes f::"'a::euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
shows "(\<integral>\<^sup>+x. f x \<partial>lborel) = integral UNIV f"
proof -
from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ereal r"
by (cases "\<integral>\<^sup>+x. f x \<partial>lborel") auto
then show ?thesis
using nn_integral_has_integral[OF f(1,2) r] by (simp add: integral_unique)
qed
lemma nn_integral_integrable_on:
fixes f::"'a::euclidean_space \<Rightarrow> real"
assumes f: "f \<in> borel_measurable borel" "\<And>x. 0 \<le> f x" "(\<integral>\<^sup>+x. f x \<partial>lborel) < \<infinity>"
shows "f integrable_on UNIV"
proof -
from f(3) obtain r where r: "(\<integral>\<^sup>+x. f x \<partial>lborel) = ereal r"
by (cases "\<integral>\<^sup>+x. f x \<partial>lborel") auto
then show ?thesis
by (intro has_integral_integrable[where i=r] nn_integral_has_integral[where r=r] f)
qed
lemma nn_integral_has_integral_lborel:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes f_borel: "f \<in> borel_measurable borel" and nonneg: "\<And>x. 0 \<le> f x"
assumes I: "(f has_integral I) UNIV"
shows "integral\<^sup>N lborel f = I"
proof -
from f_borel have "(\<lambda>x. ereal (f x)) \<in> borel_measurable lborel" by auto
from borel_measurable_implies_simple_function_sequence'[OF this] guess F . note F = this
let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
note F(1)[THEN borel_measurable_simple_function, measurable]
{ fix i x have "real (F i x) \<le> f x"
using F(3,5) F(4)[of x, symmetric] nonneg
unfolding real_le_ereal_iff
by (auto simp: image_iff eq_commute[of \<infinity>] max_def intro: SUP_upper split: split_if_asm) }
note F_le_f = this
let ?F = "\<lambda>i x. F i x * indicator (?B i) x"
have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lborel) = (SUP i. integral\<^sup>N lborel (\<lambda>x. ?F i x))"
proof (subst nn_integral_monotone_convergence_SUP[symmetric])
{ fix x
obtain j where j: "x \<in> ?B j"
using UN_box_eq_UNIV by auto
have "ereal (f x) = (SUP i. F i x)"
using F(4)[of x] nonneg[of x] by (simp add: max_def)
also have "\<dots> = (SUP i. ?F i x)"
proof (rule SUP_eq)
fix i show "\<exists>j\<in>UNIV. F i x \<le> ?F j x"
using j F(2)
by (intro bexI[of _ "max i j"])
(auto split: split_max split_indicator simp: incseq_def le_fun_def box_def)
qed (auto intro!: F split: split_indicator)
finally have "ereal (f x) = (SUP i. ?F i x)" . }
then show "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lborel) = (\<integral>\<^sup>+ x. (SUP i. ?F i x) \<partial>lborel)"
by simp
qed (insert F, auto simp: incseq_def le_fun_def box_def split: split_indicator)
also have "\<dots> \<le> ereal I"
proof (rule SUP_least)
fix i :: nat
have finite_F: "(\<integral>\<^sup>+ x. ereal (real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
proof (rule nn_integral_bound_simple_function)
have "emeasure lborel {x \<in> space lborel. ereal (real (F i x) * indicator (?B i) x) \<noteq> 0} \<le>
emeasure lborel (?B i)"
by (intro emeasure_mono) (auto split: split_indicator)
then show "emeasure lborel {x \<in> space lborel. ereal (real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
by auto
qed (auto split: split_indicator
intro!: real_of_ereal_pos F simple_function_compose1[where g="real"] simple_function_ereal)
have int_F: "(\<lambda>x. real (F i x) * indicator (?B i) x) integrable_on UNIV"
using F(5) finite_F
by (intro nn_integral_integrable_on) (auto split: split_indicator intro: real_of_ereal_pos)
have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
(\<integral>\<^sup>+ x. ereal (real (F i x) * indicator (?B i) x) \<partial>lborel)"
using F(3,5)
by (intro nn_integral_cong) (auto simp: image_iff ereal_real eq_commute split: split_indicator)
also have "\<dots> = ereal (integral UNIV (\<lambda>x. real (F i x) * indicator (?B i) x))"
using F
by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
(auto split: split_indicator intro: real_of_ereal_pos)
also have "\<dots> \<le> ereal I"
by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
split: split_indicator )
finally show "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) \<le> ereal I" .
qed
finally have "(\<integral>\<^sup>+ x. ereal (f x) \<partial>lborel) < \<infinity>"
by auto
from nn_integral_lborel_eq_integral[OF assms(1,2) this] I show ?thesis
by (simp add: integral_unique)
qed
lemma has_integral_iff_emeasure_lborel:
fixes A :: "'a::euclidean_space set"
assumes A[measurable]: "A \<in> sets borel"
shows "((\<lambda>x. 1) has_integral r) A \<longleftrightarrow> emeasure lborel A = ereal r"
proof cases
assume emeasure_A: "emeasure lborel A = \<infinity>"
have "\<not> (\<lambda>x. 1::real) integrable_on A"
proof
assume int: "(\<lambda>x. 1::real) integrable_on A"
then have "(indicator A::'a \<Rightarrow> real) integrable_on UNIV"
unfolding indicator_def[abs_def] integrable_restrict_univ .
then obtain r where "((indicator A::'a\<Rightarrow>real) has_integral r) UNIV"
by auto
from nn_integral_has_integral_lborel[OF _ _ this] emeasure_A show False
by (simp add: ereal_indicator)
qed
with emeasure_A show ?thesis
by auto
next
assume "emeasure lborel A \<noteq> \<infinity>"
moreover then have "((\<lambda>x. 1) has_integral measure lborel A) A"
by (simp add: has_integral_measure_lborel)
ultimately show ?thesis
by (auto simp: emeasure_eq_ereal_measure has_integral_unique)
qed
lemma has_integral_integral_real:
fixes f::"'a::euclidean_space \<Rightarrow> real"
assumes f: "integrable lborel f"
shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
using f proof induct
case (base A c) then show ?case
by (auto intro!: has_integral_mult_left simp: )
(simp add: emeasure_eq_ereal_measure indicator_def has_integral_measure_lborel)
next
case (add f g) then show ?case
by (auto intro!: has_integral_add)
next
case (lim f s)
show ?case
proof (rule has_integral_dominated_convergence)
show "\<And>i. (s i has_integral integral\<^sup>L lborel (s i)) UNIV" by fact
show "(\<lambda>x. norm (2 * f x)) integrable_on UNIV"
using `integrable lborel f`
by (intro nn_integral_integrable_on)
(auto simp: integrable_iff_bounded abs_mult times_ereal.simps(1)[symmetric] nn_integral_cmult
simp del: times_ereal.simps)
show "\<And>k. \<forall>x\<in>UNIV. norm (s k x) \<le> norm (2 * f x)"
using lim by (auto simp add: abs_mult)
show "\<forall>x\<in>UNIV. (\<lambda>k. s k x) ----> f x"
using lim by auto
show "(\<lambda>k. integral\<^sup>L lborel (s k)) ----> integral\<^sup>L lborel f"
using lim lim(1)[THEN borel_measurable_integrable]
by (intro integral_dominated_convergence[where w="\<lambda>x. 2 * norm (f x)"]) auto
qed
qed
context
fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
begin
lemma has_integral_integral_lborel:
assumes f: "integrable lborel f"
shows "(f has_integral (integral\<^sup>L lborel f)) UNIV"
proof -
have "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
using f by (intro has_integral_setsum finite_Basis ballI has_integral_scaleR_left has_integral_integral_real) auto
also have eq_f: "(\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) = f"
by (simp add: fun_eq_iff euclidean_representation)
also have "(\<Sum>b\<in>Basis. integral\<^sup>L lborel (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lborel f"
using f by (subst (2) eq_f[symmetric]) simp
finally show ?thesis .
qed
lemma integrable_on_lborel: "integrable lborel f \<Longrightarrow> f integrable_on UNIV"
using has_integral_integral_lborel by (auto intro: has_integral_integrable)
lemma integral_lborel: "integrable lborel f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lborel)"
using has_integral_integral_lborel by auto
end
subsection {* Fundamental Theorem of Calculus for the Lebesgue integral *}
lemma emeasure_bounded_finite:
assumes "bounded A" shows "emeasure lborel A < \<infinity>"
proof -
from bounded_subset_cbox[OF `bounded A`] obtain a b where "A \<subseteq> cbox a b"
by auto
then have "emeasure lborel A \<le> emeasure lborel (cbox a b)"
by (intro emeasure_mono) auto
then show ?thesis
by (auto simp: emeasure_lborel_cbox_eq)
qed
lemma emeasure_compact_finite: "compact A \<Longrightarrow> emeasure lborel A < \<infinity>"
using emeasure_bounded_finite[of A] by (auto intro: compact_imp_bounded)
lemma borel_integrable_compact:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes "compact S" "continuous_on S f"
shows "integrable lborel (\<lambda>x. indicator S x *\<^sub>R f x)"
proof cases
assume "S \<noteq> {}"
have "continuous_on S (\<lambda>x. norm (f x))"
using assms by (intro continuous_intros)
from continuous_attains_sup[OF `compact S` `S \<noteq> {}` this]
obtain M where M: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> M"
by auto
show ?thesis
proof (rule integrable_bound)
show "integrable lborel (\<lambda>x. indicator S x * M)"
using assms by (auto intro!: emeasure_compact_finite borel_compact integrable_mult_left)
show "(\<lambda>x. indicator S x *\<^sub>R f x) \<in> borel_measurable lborel"
using assms by (auto intro!: borel_measurable_continuous_on_indicator borel_compact)
show "AE x in lborel. norm (indicator S x *\<^sub>R f x) \<le> norm (indicator S x * M)"
by (auto split: split_indicator simp: abs_real_def dest!: M)
qed
qed simp
lemma borel_integrable_atLeastAtMost:
fixes f :: "real \<Rightarrow> real"
assumes f: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
shows "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is "integrable _ ?f")
proof -
have "integrable lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x)"
proof (rule borel_integrable_compact)
from f show "continuous_on {a..b} f"
by (auto intro: continuous_at_imp_continuous_on)
qed simp
then show ?thesis
by (auto simp: mult.commute)
qed
text {*
For the positive integral we replace continuity with Borel-measurability.
*}
lemma
fixes f :: "real \<Rightarrow> real"
assumes [measurable]: "f \<in> borel_measurable borel"
assumes f: "\<And>x. x \<in> {a..b} \<Longrightarrow> DERIV F x :> f x" "\<And>x. x \<in> {a..b} \<Longrightarrow> 0 \<le> f x" and "a \<le> b"
shows nn_integral_FTC_Icc: "(\<integral>\<^sup>+x. ereal (f x) * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?nn)
and has_bochner_integral_FTC_Icc_nonneg:
"has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
and integral_FTC_Icc_nonneg: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
and integrable_FTC_Icc_nonneg: "integrable lborel (\<lambda>x. f x * indicator {a .. b} x)" (is ?int)
proof -
have *: "(\<lambda>x. f x * indicator {a..b} x) \<in> borel_measurable borel" "\<And>x. 0 \<le> f x * indicator {a..b} x"
using f(2) by (auto split: split_indicator)
have "(f has_integral F b - F a) {a..b}"
by (intro fundamental_theorem_of_calculus)
(auto simp: has_field_derivative_iff_has_vector_derivative[symmetric]
intro: has_field_derivative_subset[OF f(1)] `a \<le> b`)
then have i: "((\<lambda>x. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
unfolding indicator_def if_distrib[where f="\<lambda>x. a * x" for a]
by (simp cong del: if_cong del: atLeastAtMost_iff)
then have nn: "(\<integral>\<^sup>+x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a"
by (rule nn_integral_has_integral_lborel[OF *])
then show ?has
by (rule has_bochner_integral_nn_integral[rotated 2]) (simp_all add: *)
then show ?eq ?int
unfolding has_bochner_integral_iff by auto
from nn show ?nn
by (simp add: ereal_mult_indicator)
qed
lemma
fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
assumes "a \<le> b"
assumes "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
assumes cont: "continuous_on {a .. b} f"
shows has_bochner_integral_FTC_Icc:
"has_bochner_integral lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) (F b - F a)" (is ?has)
and integral_FTC_Icc: "(\<integral>x. indicator {a .. b} x *\<^sub>R f x \<partial>lborel) = F b - F a" (is ?eq)
proof -
let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
have int: "integrable lborel ?f"
using borel_integrable_compact[OF _ cont] by auto
have "(f has_integral F b - F a) {a..b}"
using assms(1,2) by (intro fundamental_theorem_of_calculus) auto
moreover
have "(f has_integral integral\<^sup>L lborel ?f) {a..b}"
using has_integral_integral_lborel[OF int]
unfolding indicator_def if_distrib[where f="\<lambda>x. x *\<^sub>R a" for a]
by (simp cong del: if_cong del: atLeastAtMost_iff)
ultimately show ?eq
by (auto dest: has_integral_unique)
then show ?has
using int by (auto simp: has_bochner_integral_iff)
qed
lemma
fixes f :: "real \<Rightarrow> real"
assumes "a \<le> b"
assumes deriv: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> DERIV F x :> f x"
assumes cont: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
shows has_bochner_integral_FTC_Icc_real:
"has_bochner_integral lborel (\<lambda>x. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
and integral_FTC_Icc_real: "(\<integral>x. f x * indicator {a .. b} x \<partial>lborel) = F b - F a" (is ?eq)
proof -
have 1: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
using deriv by (auto intro: DERIV_subset)
have 2: "continuous_on {a .. b} f"
using cont by (intro continuous_at_imp_continuous_on) auto
show ?has ?eq
using has_bochner_integral_FTC_Icc[OF `a \<le> b` 1 2] integral_FTC_Icc[OF `a \<le> b` 1 2]
by (auto simp: mult.commute)
qed
lemma nn_integral_FTC_atLeast:
fixes f :: "real \<Rightarrow> real"
assumes f_borel: "f \<in> borel_measurable borel"
assumes f: "\<And>x. a \<le> x \<Longrightarrow> DERIV F x :> f x"
assumes nonneg: "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> f x"
assumes lim: "(F ---> T) at_top"
shows "(\<integral>\<^sup>+x. ereal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
proof -
let ?f = "\<lambda>(i::nat) (x::real). ereal (f x) * indicator {a..a + real i} x"
let ?fR = "\<lambda>x. ereal (f x) * indicator {a ..} x"
have "\<And>x. (SUP i::nat. ?f i x) = ?fR x"
proof (rule SUP_Lim_ereal)
show "\<And>x. incseq (\<lambda>i. ?f i x)"
using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
fix x
from reals_Archimedean2[of "x - a"] guess n ..
then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
then show "(\<lambda>n. ?f n x) ----> ?fR x"
by (rule Lim_eventually)
qed
then have "integral\<^sup>N lborel ?fR = (\<integral>\<^sup>+ x. (SUP i::nat. ?f i x) \<partial>lborel)"
by simp
also have "\<dots> = (SUP i::nat. (\<integral>\<^sup>+ x. ?f i x \<partial>lborel))"
proof (rule nn_integral_monotone_convergence_SUP)
show "incseq ?f"
using nonneg by (auto simp: incseq_def le_fun_def split: split_indicator)
show "\<And>i. (?f i) \<in> borel_measurable lborel"
using f_borel by auto
qed
also have "\<dots> = (SUP i::nat. ereal (F (a + real i) - F a))"
by (subst nn_integral_FTC_Icc[OF f_borel f nonneg]) auto
also have "\<dots> = T - F a"
proof (rule SUP_Lim_ereal)
show "incseq (\<lambda>n. ereal (F (a + real n) - F a))"
proof (simp add: incseq_def, safe)
fix m n :: nat assume "m \<le> n"
with f nonneg show "F (a + real m) \<le> F (a + real n)"
by (intro DERIV_nonneg_imp_nondecreasing[where f=F])
(simp, metis add_increasing2 order_refl order_trans real_of_nat_ge_zero)
qed
have "(\<lambda>x. F (a + real x)) ----> T"
apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
apply (rule filterlim_real_sequentially)
done
then show "(\<lambda>n. ereal (F (a + real n) - F a)) ----> ereal (T - F a)"
unfolding lim_ereal
by (intro tendsto_diff) auto
qed
finally show ?thesis .
qed
lemma integral_power:
"a \<le> b \<Longrightarrow> (\<integral>x. x^k * indicator {a..b} x \<partial>lborel) = (b^Suc k - a^Suc k) / Suc k"
proof (subst integral_FTC_Icc_real)
fix x show "DERIV (\<lambda>x. x^Suc k / Suc k) x :> x^k"
by (intro derivative_eq_intros) auto
qed (auto simp: field_simps)
subsection {* Integration by parts *}
lemma integral_by_parts_integrable:
fixes f g F G::"real \<Rightarrow> real"
assumes "a \<le> b"
assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
assumes [intro]: "!!x. DERIV F x :> f x"
assumes [intro]: "!!x. DERIV G x :> g x"
shows "integrable lborel (\<lambda>x.((F x) * (g x) + (f x) * (G x)) * indicator {a .. b} x)"
by (auto intro!: borel_integrable_atLeastAtMost continuous_intros) (auto intro!: DERIV_isCont)
lemma integral_by_parts:
fixes f g F G::"real \<Rightarrow> real"
assumes [arith]: "a \<le> b"
assumes cont_f[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
assumes cont_g[intro]: "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
assumes [intro]: "!!x. DERIV F x :> f x"
assumes [intro]: "!!x. DERIV G x :> g x"
shows "(\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel)
= F b * G b - F a * G a - \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
proof-
have 0: "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) = F b * G b - F a * G a"
by (rule integral_FTC_Icc_real, auto intro!: derivative_eq_intros continuous_intros)
(auto intro!: DERIV_isCont)
have "(\<integral>x. (F x * g x + f x * G x) * indicator {a .. b} x \<partial>lborel) =
(\<integral>x. (F x * g x) * indicator {a .. b} x \<partial>lborel) + \<integral>x. (f x * G x) * indicator {a .. b} x \<partial>lborel"
apply (subst integral_add[symmetric])
apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
by (auto intro!: DERIV_isCont integral_cong split:split_indicator)
thus ?thesis using 0 by auto
qed
lemma integral_by_parts':
fixes f g F G::"real \<Rightarrow> real"
assumes "a \<le> b"
assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont f x"
assumes "!!x. a \<le>x \<Longrightarrow> x\<le>b \<Longrightarrow> isCont g x"
assumes "!!x. DERIV F x :> f x"
assumes "!!x. DERIV G x :> g x"
shows "(\<integral>x. indicator {a .. b} x *\<^sub>R (F x * g x) \<partial>lborel)
= F b * G b - F a * G a - \<integral>x. indicator {a .. b} x *\<^sub>R (f x * G x) \<partial>lborel"
using integral_by_parts[OF assms] by (simp add: ac_simps)
lemma has_bochner_integral_even_function:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
assumes even: "\<And>x. f (- x) = f x"
shows "has_bochner_integral lborel f (2 *\<^sub>R x)"
proof -
have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
by (auto split: split_indicator)
have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) x"
by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
(auto simp: indicator even f)
with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + x)"
by (rule has_bochner_integral_add)
then have "has_bochner_integral lborel f (x + x)"
by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
(auto split: split_indicator)
then show ?thesis
by (simp add: scaleR_2)
qed
lemma has_bochner_integral_odd_function:
fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
assumes f: "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x) x"
assumes odd: "\<And>x. f (- x) = - f x"
shows "has_bochner_integral lborel f 0"
proof -
have indicator: "\<And>x::real. indicator {..0} (- x) = indicator {0..} x"
by (auto split: split_indicator)
have "has_bochner_integral lborel (\<lambda>x. - indicator {.. 0} x *\<^sub>R f x) x"
by (subst lborel_has_bochner_integral_real_affine_iff[where c="-1" and t=0])
(auto simp: indicator odd f)
from has_bochner_integral_minus[OF this]
have "has_bochner_integral lborel (\<lambda>x. indicator {.. 0} x *\<^sub>R f x) (- x)"
by simp
with f have "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R f x + indicator {.. 0} x *\<^sub>R f x) (x + - x)"
by (rule has_bochner_integral_add)
then have "has_bochner_integral lborel f (x + - x)"
by (rule has_bochner_integral_discrete_difference[where X="{0}", THEN iffD1, rotated 4])
(auto split: split_indicator)
then show ?thesis
by simp
qed
end