prove HK-integrable implies Lebesgue measurable; prove HK-integral equals Lebesgue integral for nonneg functions
theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure
begin
lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
by (auto intro: order_trans)
lemma ball_trans:
assumes "y \<in> ball z q" "r + q \<le> s" shows "ball y r \<subseteq> ball z s"
proof safe
fix x assume x: "x \<in> ball y r"
have "dist z x \<le> dist z y + dist y x"
by (rule dist_triangle)
also have "\<dots> < s"
using assms x by auto
finally show "x \<in> ball z s"
by simp
qed
abbreviation lebesgue :: "'a::euclidean_space measure"
where "lebesgue \<equiv> completion lborel"
abbreviation lebesgue_on :: "'a set \<Rightarrow> 'a::euclidean_space measure"
where "lebesgue_on \<Omega> \<equiv> restrict_space (completion lborel) \<Omega>"
lemma has_integral_implies_lebesgue_measurable_cbox:
fixes f :: "'a :: euclidean_space \<Rightarrow> real"
assumes f: "(f has_integral I) (cbox x y)"
shows "f \<in> lebesgue_on (cbox x y) \<rightarrow>\<^sub>M borel"
proof (rule cld_measure.borel_measurable_cld)
let ?L = "lebesgue_on (cbox x y)"
let ?\<mu> = "emeasure ?L"
let ?\<mu>' = "outer_measure_of ?L"
interpret L: finite_measure ?L
proof
show "?\<mu> (space ?L) \<noteq> \<infinity>"
by (simp add: emeasure_restrict_space space_restrict_space emeasure_lborel_cbox_eq)
qed
show "cld_measure ?L"
proof
fix B A assume "B \<subseteq> A" "A \<in> null_sets ?L"
then show "B \<in> sets ?L"
using null_sets_completion_subset[OF \<open>B \<subseteq> A\<close>, of lborel]
by (auto simp add: null_sets_restrict_space sets_restrict_space_iff intro: )
next
fix A assume "A \<subseteq> space ?L" "\<And>B. B \<in> sets ?L \<Longrightarrow> ?\<mu> B < \<infinity> \<Longrightarrow> A \<inter> B \<in> sets ?L"
from this(1) this(2)[of "space ?L"] show "A \<in> sets ?L"
by (auto simp: Int_absorb2 less_top[symmetric])
qed auto
then interpret cld_measure ?L
.
have content_eq_L: "A \<in> sets borel \<Longrightarrow> A \<subseteq> cbox x y \<Longrightarrow> content A = measure ?L A" for A
by (subst measure_restrict_space) (auto simp: measure_def)
fix E and a b :: real assume "E \<in> sets ?L" "a < b" "0 < ?\<mu> E" "?\<mu> E < \<infinity>"
then obtain M :: real where "?\<mu> E = M" "0 < M"
by (cases "?\<mu> E") auto
define e where "e = M / (4 + 2 / (b - a))"
from \<open>a < b\<close> \<open>0<M\<close> have "0 < e"
by (auto intro!: divide_pos_pos simp: field_simps e_def)
have "e < M / (3 + 2 / (b - a))"
using \<open>a < b\<close> \<open>0 < M\<close>
unfolding e_def by (intro divide_strict_left_mono add_strict_right_mono mult_pos_pos) (auto simp: field_simps)
then have "2 * e < (b - a) * (M - e * 3)"
using \<open>0<M\<close> \<open>0 < e\<close> \<open>a < b\<close> by (simp add: field_simps)
have e_less_M: "e < M / 1"
unfolding e_def using \<open>a < b\<close> \<open>0<M\<close> by (intro divide_strict_left_mono) (auto simp: field_simps)
obtain d
where "gauge d"
and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
using \<open>0<e\<close> f unfolding has_integral by auto
define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
have "incseq (C X)" for X
unfolding C_def [abs_def]
by (intro monoI Collect_mono conj_mono imp_refl le_left_mono subset_ball divide_left_mono Int_mono) auto
{ fix X assume "X \<subseteq> space ?L" and eq: "?\<mu>' X = ?\<mu> E"
have "(SUP m. outer_measure_of ?L (C X m)) = outer_measure_of ?L (\<Union>m. C X m)"
using \<open>X \<subseteq> space ?L\<close> by (intro SUP_outer_measure_of_incseq \<open>incseq (C X)\<close>) (auto simp: C_def)
also have "(\<Union>m. C X m) = X"
proof -
{ fix x
obtain e where "0 < e" "ball x e \<subseteq> d x"
using gaugeD[OF \<open>gauge d\<close>, of x] unfolding open_contains_ball by auto
moreover
obtain n where "1 / (1 + real n) < e"
using reals_Archimedean[OF \<open>0<e\<close>] by (auto simp: inverse_eq_divide)
then have "ball x (1 / (1 + real n)) \<subseteq> ball x e"
by (intro subset_ball) auto
ultimately have "\<exists>n. ball x (1 / (1 + real n)) \<subseteq> d x"
by blast }
then show ?thesis
by (auto simp: C_def)
qed
finally have "(SUP m. outer_measure_of ?L (C X m)) = ?\<mu> E"
using eq by auto
also have "\<dots> > M - e"
using \<open>0 < M\<close> \<open>?\<mu> E = M\<close> \<open>0<e\<close> by (auto intro!: ennreal_lessI)
finally have "\<exists>m. M - e < outer_measure_of ?L (C X m)"
unfolding less_SUP_iff by auto }
note C = this
let ?E = "{x\<in>E. f x \<le> a}" and ?F = "{x\<in>E. b \<le> f x}"
have "\<not> (?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E)"
proof
assume eq: "?\<mu>' ?E = ?\<mu> E \<and> ?\<mu>' ?F = ?\<mu> E"
with C[of ?E] C[of ?F] \<open>E \<in> sets ?L\<close>[THEN sets.sets_into_space] obtain ma mb
where "M - e < outer_measure_of ?L (C ?E ma)" "M - e < outer_measure_of ?L (C ?F mb)"
by auto
moreover define m where "m = max ma mb"
ultimately have M_minus_e: "M - e < outer_measure_of ?L (C ?E m)" "M - e < outer_measure_of ?L (C ?F m)"
using
incseqD[OF \<open>incseq (C ?E)\<close>, of ma m, THEN outer_measure_of_mono]
incseqD[OF \<open>incseq (C ?F)\<close>, of mb m, THEN outer_measure_of_mono]
by (auto intro: less_le_trans)
define d' where "d' x = d x \<inter> ball x (1 / (3 * Suc m))" for x
have "gauge d'"
unfolding d'_def by (intro gauge_inter \<open>gauge d\<close> gauge_ball) auto
then obtain p where p: "p tagged_division_of cbox x y" "d' fine p"
by (rule fine_division_exists)
then have "d fine p"
unfolding d'_def[abs_def] fine_def by auto
define s where "s = {(x::'a, k). k \<inter> (C ?E m) \<noteq> {} \<and> k \<inter> (C ?F m) \<noteq> {}}"
define T where "T E k = (SOME x. x \<in> k \<inter> C E m)" for E k
let ?A = "(\<lambda>(x, k). (T ?E k, k)) ` (p \<inter> s) \<union> (p - s)"
let ?B = "(\<lambda>(x, k). (T ?F k, k)) ` (p \<inter> s) \<union> (p - s)"
{ fix X assume X_eq: "X = ?E \<or> X = ?F"
let ?T = "(\<lambda>(x, k). (T X k, k))"
let ?p = "?T ` (p \<inter> s) \<union> (p - s)"
have in_s: "(x, k) \<in> s \<Longrightarrow> T X k \<in> k \<inter> C X m" for x k
using someI_ex[of "\<lambda>x. x \<in> k \<inter> C X m"] X_eq unfolding ex_in_conv by (auto simp: T_def s_def)
{ fix x k assume "(x, k) \<in> p" "(x, k) \<in> s"
have k: "k \<subseteq> ball x (1 / (3 * Suc m))"
using \<open>d' fine p\<close>[THEN fineD, OF \<open>(x, k) \<in> p\<close>] by (auto simp: d'_def)
then have "x \<in> ball (T X k) (1 / (3 * Suc m))"
using in_s[OF \<open>(x, k) \<in> s\<close>] by (auto simp: C_def subset_eq dist_commute)
then have "ball x (1 / (3 * Suc m)) \<subseteq> ball (T X k) (1 / Suc m)"
by (rule ball_trans) (auto simp: divide_simps)
with k in_s[OF \<open>(x, k) \<in> s\<close>] have "k \<subseteq> d (T X k)"
by (auto simp: C_def) }
then have "d fine ?p"
using \<open>d fine p\<close> by (auto intro!: fineI)
moreover
have "?p tagged_division_of cbox x y"
proof (rule tagged_division_ofI)
show "finite ?p"
using p(1) by auto
next
fix z k assume *: "(z, k) \<in> ?p"
then consider "(z, k) \<in> p" "(z, k) \<notin> s"
| x' where "(x', k) \<in> p" "(x', k) \<in> s" "z = T X k"
by (auto simp: T_def)
then have "z \<in> k \<and> k \<subseteq> cbox x y \<and> (\<exists>a b. k = cbox a b)"
using p(1) by cases (auto dest: in_s)
then show "z \<in> k" "k \<subseteq> cbox x y" "\<exists>a b. k = cbox a b"
by auto
next
fix z k z' k' assume "(z, k) \<in> ?p" "(z', k') \<in> ?p" "(z, k) \<noteq> (z', k')"
with tagged_division_ofD(5)[OF p(1), of _ k _ k']
show "interior k \<inter> interior k' = {}"
by (auto simp: T_def dest: in_s)
next
have "{k. \<exists>x. (x, k) \<in> ?p} = {k. \<exists>x. (x, k) \<in> p}"
by (auto simp: T_def image_iff Bex_def)
then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
using p(1) by auto
qed
ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e"
using integral_f by auto
have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
using p(1)[THEN tagged_division_ofD(1)]
by (safe intro!: setsum.union_inter_neutral) (auto simp: s_def T_def)
also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
proof (subst setsum.reindex_nontrivial, safe)
fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
with tagged_division_ofD(5)[OF p(1), of x1 k x2 k] tagged_division_ofD(4)[OF p(1), of x1 k]
show "x1 = x2"
by (auto simp: content_eq_0_interior)
qed (use p in \<open>auto intro!: setsum.cong\<close>)
finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
(\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
using in_s[of x k] by (auto simp: C_def)
note I eq in_T }
note parts = this
have p_in_L: "(x, k) \<in> p \<Longrightarrow> k \<in> sets ?L" for x k
using tagged_division_ofD(3, 4)[OF p(1), of x k] by (auto simp: sets_restrict_space)
have [simp]: "finite p"
using tagged_division_ofD(1)[OF p(1)] .
have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)"
proof (intro mult_right_mono)
have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
have sets: "(E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<in> sets ?L" for X
using tagged_division_ofD(1)[OF p(1)] by (intro sets.Diff \<open>E \<in> sets ?L\<close> sets.finite_Union sets.Int) (auto intro: p_in_L)
{ fix X assume "X \<subseteq> E" "M - e < ?\<mu>' (C X m)"
have "M - e \<le> ?\<mu>' (C X m)"
by (rule less_imp_le) fact
also have "\<dots> \<le> ?\<mu>' (E - (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}))"
proof (intro outer_measure_of_mono subsetI)
fix v assume "v \<in> C X m"
then have "v \<in> cbox x y" "v \<in> E"
using \<open>E \<subseteq> space ?L\<close> \<open>X \<subseteq> E\<close> by (auto simp: space_restrict_space C_def)
then obtain z k where "(z, k) \<in> p" "v \<in> k"
using tagged_division_ofD(6)[OF p(1), symmetric] by auto
then show "v \<in> E - E \<inter> (\<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
using \<open>v \<in> C X m\<close> \<open>v \<in> E\<close> by auto
qed
also have "\<dots> = ?\<mu> E - ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})"
using \<open>E \<in> sets ?L\<close> fin[of X] sets[of X] by (auto intro!: emeasure_Diff)
finally have "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) \<le> e"
using \<open>0 < e\<close> e_less_M apply (cases "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}})")
by (auto simp add: \<open>?\<mu> E = M\<close> ennreal_minus ennreal_le_iff2)
note this }
note upper_bound = this
have "?\<mu> (E \<inter> \<Union>(snd`(p - s))) =
?\<mu> ((E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) \<union> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}}))"
by (intro arg_cong[where f="?\<mu>"]) (auto simp: s_def image_def Bex_def)
also have "\<dots> \<le> ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?E m = {}}) + ?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C ?F m = {}})"
using sets[of ?E] sets[of ?F] M_minus_e by (intro emeasure_subadditive) auto
also have "\<dots> \<le> e + ennreal e"
using upper_bound[of ?E] upper_bound[of ?F] M_minus_e by (intro add_mono) auto
finally have "?\<mu> E - 2*e \<le> ?\<mu> (E - (E \<inter> \<Union>(snd`(p - s))))"
using \<open>0 < e\<close> \<open>E \<in> sets ?L\<close> tagged_division_ofD(1)[OF p(1)]
by (subst emeasure_Diff)
(auto simp: ennreal_plus[symmetric] top_unique simp del: ennreal_plus
intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
also have "\<dots> \<le> ?\<mu> (\<Union>x\<in>p \<inter> s. snd x)"
proof (safe intro!: emeasure_mono subsetI)
fix v assume "v \<in> E" and not: "v \<notin> (\<Union>x\<in>p \<inter> s. snd x)"
then have "v \<in> cbox x y"
using \<open>E \<subseteq> space ?L\<close> by (auto simp: space_restrict_space)
then obtain z k where "(z, k) \<in> p" "v \<in> k"
using tagged_division_ofD(6)[OF p(1), symmetric] by auto
with not show "v \<in> UNION (p - s) snd"
by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"])
qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
also have "\<dots> = measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
by (auto intro!: emeasure_eq_ennreal_measure)
finally have "M - 2 * e \<le> measure ?L (\<Union>x\<in>p \<inter> s. snd x)"
unfolding \<open>?\<mu> E = M\<close> using \<open>0 < e\<close> by (simp add: ennreal_minus)
also have "measure ?L (\<Union>x\<in>p \<inter> s. snd x) = content (\<Union>x\<in>p \<inter> s. snd x)"
using tagged_division_ofD(1,3,4) [OF p(1)]
by (intro content_eq_L[symmetric])
(fastforce intro!: sets.finite_UN UN_least del: subsetI)+
also have "content (\<Union>x\<in>p \<inter> s. snd x) \<le> (\<Sum>k\<in>p \<inter> s. content (snd k))"
using p(1) by (auto simp: emeasure_lborel_cbox_eq intro!: measure_subadditive_finite
dest!: p(1)[THEN tagged_division_ofD(4)])
finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
using \<open>0 < e\<close> by (simp add: split_beta)
qed (use \<open>a < b\<close> in auto)
also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
by (simp add: setsum_distrib_right split_beta')
also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
using parts(3) by (auto intro!: setsum_mono mult_left_mono diff_mono)
also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
by (auto intro!: setsum.cong simp: field_simps setsum_subtractf[symmetric])
also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
by (subst (1 2) parts) auto
also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
by auto
also have "\<dots> \<le> e + e"
using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
finally show False
using \<open>2 * e < (b - a) * (M - e * 3)\<close> by (auto simp: field_simps)
qed
moreover have "?\<mu>' ?E \<le> ?\<mu> E" "?\<mu>' ?F \<le> ?\<mu> E"
unfolding outer_measure_of_eq[OF \<open>E \<in> sets ?L\<close>, symmetric] by (auto intro!: outer_measure_of_mono)
ultimately show "min (?\<mu>' ?E) (?\<mu>' ?F) < ?\<mu> E"
unfolding min_less_iff_disj by (auto simp: less_le)
qed
lemma has_integral_implies_lebesgue_measurable_real:
fixes f :: "'a :: euclidean_space \<Rightarrow> real"
assumes f: "(f has_integral I) \<Omega>"
shows "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
proof -
define B :: "nat \<Rightarrow> 'a set" where "B n = cbox (- real n *\<^sub>R One) (real n *\<^sub>R One)" for n
show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
proof (rule measurable_piecewise_restrict)
have "(\<Union>n. box (- real n *\<^sub>R One) (real n *\<^sub>R One)) \<subseteq> UNION UNIV B"
unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
then show "countable (range B)" "space lebesgue \<subseteq> UNION UNIV B"
by (auto simp: B_def UN_box_eq_UNIV)
next
fix \<Omega>' assume "\<Omega>' \<in> range B"
then obtain n where \<Omega>': "\<Omega>' = B n" by auto
then show "\<Omega>' \<inter> space lebesgue \<in> sets lebesgue"
by (auto simp: B_def)
have "f integrable_on \<Omega>"
using f by auto
then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on \<Omega>"
by (auto simp: integrable_on_def cong: has_integral_cong)
then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on (\<Omega> \<union> B n)"
by (rule integrable_on_superset[rotated 2]) auto
then have "(\<lambda>x. f x * indicator \<Omega> x) integrable_on B n"
unfolding B_def by (rule integrable_on_subcbox) auto
then show "(\<lambda>x. f x * indicator \<Omega> x) \<in> lebesgue_on \<Omega>' \<rightarrow>\<^sub>M borel"
unfolding B_def \<Omega>' by (auto intro: has_integral_implies_lebesgue_measurable_cbox simp: integrable_on_def)
qed
qed
lemma has_integral_implies_lebesgue_measurable:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes f: "(f has_integral I) \<Omega>"
shows "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
proof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI)
fix i :: "'b" assume "i \<in> Basis"
have "(\<lambda>x. (f x \<bullet> i) * indicator \<Omega> x) \<in> borel_measurable (completion lborel)"
using has_integral_linear[OF f bounded_linear_inner_left, of i]
by (intro has_integral_implies_lebesgue_measurable_real) (auto simp: comp_def)
then show "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x \<bullet> i) \<in> borel_measurable (completion lborel)"
by (simp add: ac_simps)
qed
subsection \<open>Equivalence Lebesgue integral on @{const lborel} and HK-integral\<close>
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_cong[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) \<longlonglongrightarrow> (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
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)) \<longlonglongrightarrow> ?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 top_unique)
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) \<longlonglongrightarrow> 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) \<longlonglongrightarrow> 1"
by (simp add: indicator_def UN_box_eq_UNIV) }
have "(\<lambda>n. emeasure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> 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_ennreal_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> top"
by (auto simp: top_unique)
qed
finally show "(\<lambda>n. measure lborel (A \<inter> ?B n)) \<longlonglongrightarrow> measure lborel A"
using emeasure_eq_ennreal_measure[of lborel A] finite
by (simp add: UN_box_eq_UNIV less_top)
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) = ennreal r" "0 \<le> r"
shows "(f has_integral r) UNIV"
using f proof (induct f arbitrary: r rule: borel_measurable_induct_real)
case (set A)
then have "((\<lambda>x. 1) has_integral measure lborel A) A"
by (intro has_integral_measure_lborel) (auto simp: ennreal_indicator)
with set show ?case
by (simp add: ennreal_indicator measure_def) (simp add: indicator_def)
next
case (mult g c)
then have "ennreal c * (\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal r"
by (subst nn_integral_cmult[symmetric]) (auto simp: ennreal_mult)
with \<open>0 \<le> r\<close> \<open>0 \<le> c\<close>
obtain r' where "(c = 0 \<and> r = 0) \<or> (0 \<le> r' \<and> (\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel) = ennreal r' \<and> r = c * r')"
by (cases "\<integral>\<^sup>+ x. ennreal (g x) \<partial>lborel" rule: ennreal_cases)
(auto split: if_split_asm simp: ennreal_mult_top ennreal_mult[symmetric])
with mult show ?case
by (auto intro!: has_integral_cmult_real)
next
case (add g h)
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)"
by (simp add: nn_integral_add)
with add obtain a b where "0 \<le> a" "0 \<le> b" "(\<integral>\<^sup>+ x. h x \<partial>lborel) = ennreal a" "(\<integral>\<^sup>+ x. g x \<partial>lborel) = ennreal b" "r = a + b"
by (cases "\<integral>\<^sup>+ x. h x \<partial>lborel" "\<integral>\<^sup>+ x. g x \<partial>lborel" rule: ennreal2_cases)
(auto simp: add_top nn_integral_add top_add ennreal_plus[symmetric] simp del: ennreal_plus)
with add 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. U i x \<partial>lborel) \<le> (\<integral>\<^sup>+x. f x \<partial>lborel)"
using seq(2) f(2) U_le_f by (intro nn_integral_mono) simp
then obtain p where "(\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal p" "p \<le> r" "0 \<le> p"
using seq(6) \<open>0\<le>r\<close> by (cases "\<integral>\<^sup>+x. U i x \<partial>lborel" rule: ennreal_cases) (auto simp: top_unique)
moreover note seq
ultimately have "\<exists>p. (\<integral>\<^sup>+x. U i x \<partial>lborel) = ennreal 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. ennreal (U i x) \<partial>lborel) = ennreal (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)) \<longlonglongrightarrow> 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 \<open>incseq U\<close> 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) \<longlonglongrightarrow> f x"
using seq by auto
qed
moreover have "(\<lambda>i. (\<integral>\<^sup>+x. U i x \<partial>lborel)) \<longlonglongrightarrow> (\<integral>\<^sup>+x. f x \<partial>lborel)"
using seq f(2) U_le_f by (intro nn_integral_dominated_convergence[where w=f]) auto
ultimately have "integral UNIV f = r"
by (auto simp add: bnd 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) = ennreal r" "0 \<le> r"
by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) 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) = ennreal r" "0 \<le> r"
by (cases "\<integral>\<^sup>+x. f x \<partial>lborel" rule: ennreal_cases) 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. ennreal (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]
have "0 \<le> I"
using I by (rule has_integral_nonneg) (simp add: nonneg)
have F_le_f: "enn2real (F i x) \<le> f x" for i x
using F(3,4)[where x=x] nonneg SUP_upper[of i UNIV "\<lambda>i. F i x"]
by (cases "F i x" rule: ennreal_cases) auto
let ?F = "\<lambda>i x. F i x * indicator (?B i) x"
have "(\<integral>\<^sup>+ x. ennreal (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 "ennreal (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 "ennreal (f x) = (SUP i. ?F i x)" . }
then show "(\<integral>\<^sup>+ x. ennreal (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> ennreal I"
proof (rule SUP_least)
fix i :: nat
have finite_F: "(\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel) < \<infinity>"
proof (rule nn_integral_bound_simple_function)
have "emeasure lborel {x \<in> space lborel. ennreal (enn2real (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. ennreal (enn2real (F i x) * indicator (?B i) x) \<noteq> 0} < \<infinity>"
by (auto simp: less_top[symmetric] top_unique)
qed (auto split: split_indicator
intro!: F simple_function_compose1[where g="enn2real"] simple_function_ennreal)
have int_F: "(\<lambda>x. enn2real (F i x) * indicator (?B i) x) integrable_on UNIV"
using F(4) finite_F
by (intro nn_integral_integrable_on) (auto split: split_indicator simp: enn2real_nonneg)
have "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) =
(\<integral>\<^sup>+ x. ennreal (enn2real (F i x) * indicator (?B i) x) \<partial>lborel)"
using F(3,4)
by (intro nn_integral_cong) (auto simp: image_iff eq_commute split: split_indicator)
also have "\<dots> = ennreal (integral UNIV (\<lambda>x. enn2real (F i x) * indicator (?B i) x))"
using F
by (intro nn_integral_lborel_eq_integral[OF _ _ finite_F])
(auto split: split_indicator intro: enn2real_nonneg)
also have "\<dots> \<le> ennreal I"
by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
simp: \<open>0 \<le> I\<close> split: split_indicator )
finally show "(\<integral>\<^sup>+ x. F i x * indicator (?B i) x \<partial>lborel) \<le> ennreal I" .
qed
finally have "(\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel) < \<infinity>"
by (auto simp: less_top[symmetric] top_unique)
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" and [simp]: "0 \<le> r"
shows "((\<lambda>x. 1) has_integral r) A \<longleftrightarrow> emeasure lborel A = ennreal r"
proof (cases "emeasure lborel A = \<infinity>")
case emeasure_A: True
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: ennreal_indicator)
qed
with emeasure_A show ?thesis
by auto
next
case False
then have "((\<lambda>x. 1) has_integral measure lborel A) A"
by (simp add: has_integral_measure_lborel less_top)
with False show ?thesis
by (auto simp: emeasure_eq_ennreal_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_ennreal_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 \<open>integrable lborel f\<close>
by (intro nn_integral_integrable_on)
(auto simp: integrable_iff_bounded abs_mult nn_integral_cmult ennreal_mult ennreal_mult_less_top)
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) \<longlonglongrightarrow> f x"
using lim by auto
show "(\<lambda>k. integral\<^sup>L lborel (s k)) \<longlonglongrightarrow> 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
lemma has_integral_AE:
assumes ae: "AE x in lborel. x \<in> \<Omega> \<longrightarrow> f x = g x"
shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
proof -
from ae obtain N
where N: "N \<in> sets borel" "emeasure lborel N = 0" "{x. \<not> (x \<in> \<Omega> \<longrightarrow> f x = g x)} \<subseteq> N"
by (auto elim!: AE_E)
then have not_N: "AE x in lborel. x \<notin> N"
by (simp add: AE_iff_measurable)
show ?thesis
proof (rule has_integral_spike_eq[symmetric])
show "\<forall>x\<in>\<Omega> - N. f x = g x" using N(3) by auto
show "negligible N"
unfolding negligible_def
proof (intro allI)
fix a b :: "'a"
let ?F = "\<lambda>x::'a. if x \<in> cbox a b then indicator N x else 0 :: real"
have "integrable lborel ?F = integrable lborel (\<lambda>x::'a. 0::real)"
using not_N N(1) by (intro integrable_cong_AE) auto
moreover have "(LINT x|lborel. ?F x) = (LINT x::'a|lborel. 0::real)"
using not_N N(1) by (intro integral_cong_AE) auto
ultimately have "(?F has_integral 0) UNIV"
using has_integral_integral_real[of ?F] by simp
then show "(indicator N has_integral (0::real)) (cbox a b)"
unfolding has_integral_restrict_univ .
qed
qed
qed
lemma nn_integral_has_integral_lebesgue:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes nonneg: "\<And>x. 0 \<le> f x" and I: "(f has_integral I) \<Omega>"
shows "integral\<^sup>N lborel (\<lambda>x. indicator \<Omega> x * f x) = I"
proof -
from I have "(\<lambda>x. indicator \<Omega> x *\<^sub>R f x) \<in> lebesgue \<rightarrow>\<^sub>M borel"
by (rule has_integral_implies_lebesgue_measurable)
then obtain f' :: "'a \<Rightarrow> real"
where [measurable]: "f' \<in> borel \<rightarrow>\<^sub>M borel" and eq: "AE x in lborel. indicator \<Omega> x * f x = f' x"
by (auto dest: completion_ex_borel_measurable_real)
from I have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV"
using nonneg by (simp add: indicator_def if_distrib[of "\<lambda>x. x * f y" for y] cong: if_cong)
also have "((\<lambda>x. abs (indicator \<Omega> x * f x)) has_integral I) UNIV \<longleftrightarrow> ((\<lambda>x. abs (f' x)) has_integral I) UNIV"
using eq by (intro has_integral_AE) auto
finally have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = I"
by (rule nn_integral_has_integral_lborel[rotated 2]) auto
also have "integral\<^sup>N lborel (\<lambda>x. abs (f' x)) = integral\<^sup>N lborel (\<lambda>x. abs (indicator \<Omega> x * f x))"
using eq by (intro nn_integral_cong_AE) auto
finally show ?thesis
using nonneg by auto
qed
lemma has_integral_iff_nn_integral_lebesgue:
assumes f: "\<And>x. 0 \<le> f x"
shows "(f has_integral r) UNIV \<longleftrightarrow> (f \<in> lebesgue \<rightarrow>\<^sub>M borel \<and> integral\<^sup>N lebesgue f = r \<and> 0 \<le> r)" (is "?I = ?N")
proof
assume ?I
have "0 \<le> r"
using has_integral_nonneg[OF \<open>?I\<close>] f by auto
then show ?N
using nn_integral_has_integral_lebesgue[OF f \<open>?I\<close>]
has_integral_implies_lebesgue_measurable[OF \<open>?I\<close>]
by (auto simp: nn_integral_completion)
next
assume ?N
then obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x"
by (auto dest: completion_ex_borel_measurable_real)
moreover have "(\<integral>\<^sup>+ x. ennreal \<bar>f' x\<bar> \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal \<bar>f x\<bar> \<partial>lborel)"
using f' by (intro nn_integral_cong_AE) auto
moreover have "((\<lambda>x. \<bar>f' x\<bar>) has_integral r) UNIV \<longleftrightarrow> ((\<lambda>x. \<bar>f x\<bar>) has_integral r) UNIV"
using f' by (intro has_integral_AE) auto
moreover note nn_integral_has_integral[of "\<lambda>x. \<bar>f' x\<bar>" r] \<open>?N\<close>
ultimately show ?I
using f by (auto simp: nn_integral_completion)
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
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 \<open>Fundamental Theorem of Calculus for the Lebesgue integral\<close>
text \<open>
For the positive integral we replace continuity with Borel-measurability.
\<close>
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. ennreal (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_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> b\<Longrightarrow> F x \<le> F y" for x y
using f by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
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)] \<open>a \<le> b\<close>)
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_weak_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 3]) (simp_all add: * F_mono \<open>a \<le> b\<close>)
then show ?eq ?int
unfolding has_bochner_integral_iff by auto
show ?nn
by (subst nn[symmetric])
(auto intro!: nn_integral_cong simp add: ennreal_mult f split: split_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_weak_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 \<open>a \<le> b\<close> 1 2] integral_FTC_Icc[OF \<open>a \<le> b\<close> 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 \<longlongrightarrow> T) at_top"
shows "(\<integral>\<^sup>+x. ennreal (f x) * indicator {a ..} x \<partial>lborel) = T - F a"
proof -
let ?f = "\<lambda>(i::nat) (x::real). ennreal (f x) * indicator {a..a + real i} x"
let ?fR = "\<lambda>x. ennreal (f x) * indicator {a ..} x"
have F_mono: "a \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> F x \<le> F y" for x y
using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
by (intro tendsto_le_const[OF _ lim])
(auto simp: trivial_limit_at_top_linorder eventually_at_top_linorder)
have "(SUP i::nat. ?f i x) = ?fR x" for x
proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
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) \<longlonglongrightarrow> ?fR x"
by (rule Lim_eventually)
qed (auto simp: nonneg incseq_def le_fun_def split: split_indicator)
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. ennreal (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 LIMSEQ_unique[OF LIMSEQ_SUP])
have "(\<lambda>x. F (a + real x)) \<longlonglongrightarrow> 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. ennreal (F (a + real n) - F a)) \<longlonglongrightarrow> ennreal (T - F a)"
by (simp add: F_mono F_le_T tendsto_diff)
qed (auto simp: incseq_def intro!: ennreal_le_iff[THEN iffD2] F_mono)
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 simp del: of_nat_Suc)
subsection \<open>Integration by parts\<close>
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 Bochner_Integration.integral_add[symmetric])
apply (auto intro!: borel_integrable_atLeastAtMost continuous_intros)
by (auto intro!: DERIV_isCont Bochner_Integration.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