Removed more "guess", etc.
(* Title: HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
*)
theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
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
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_Int \<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!: sum.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 sum.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!: sum.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: sum_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!: sum_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!: sum.cong simp: field_simps sum_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_diff)
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_sum simp del: Int_iff)
show "\<And>k x. ?f k x \<le> ?f (Suc k) x"
by (intro sum_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: * sum.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 ennreal_max_0: "ennreal (max 0 x) = ennreal x"
by (auto simp: max_def ennreal_neg)
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"
proof -
from integrableE[OF f] obtain r q
where "0 \<le> r" "0 \<le> q"
and r: "(\<integral>\<^sup>+ x. ennreal (max 0 (f x)) \<partial>lborel) = ennreal r"
and q: "(\<integral>\<^sup>+ x. ennreal (max 0 (- f x)) \<partial>lborel) = ennreal q"
and f: "f \<in> borel_measurable lborel" and eq: "integral\<^sup>L lborel f = r - q"
unfolding ennreal_max_0 by auto
then have "((\<lambda>x. max 0 (f x)) has_integral r) UNIV" "((\<lambda>x. max 0 (- f x)) has_integral q) UNIV"
using nn_integral_has_integral[OF _ _ r] nn_integral_has_integral[OF _ _ q] by auto
note has_integral_diff[OF this]
moreover have "(\<lambda>x. max 0 (f x) - max 0 (- f x)) = f"
by auto
ultimately show ?thesis
by (simp add: eq)
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 "\<And>x. x\<in>\<Omega> - N \<Longrightarrow> 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_sum 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
context
begin
private lemma has_integral_integral_lebesgue_real:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes f: "integrable lebesgue f"
shows "(f has_integral (integral\<^sup>L lebesgue f)) UNIV"
proof -
obtain f' where f': "f' \<in> borel \<rightarrow>\<^sub>M borel" "AE x in lborel. f x = f' x"
using completion_ex_borel_measurable_real[OF borel_measurable_integrable[OF f]] by auto
moreover have "(\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>lborel) = (\<integral>\<^sup>+ x. ennreal (norm (f' x)) \<partial>lborel)"
using f' by (intro nn_integral_cong_AE) auto
ultimately have "integrable lborel f'"
using f by (auto simp: integrable_iff_bounded nn_integral_completion cong: nn_integral_cong_AE)
note has_integral_integral_real[OF this]
moreover have "integral\<^sup>L lebesgue f = integral\<^sup>L lebesgue f'"
using f' f by (intro integral_cong_AE) (auto intro: AE_completion measurable_completion)
moreover have "integral\<^sup>L lebesgue f' = integral\<^sup>L lborel f'"
using f' by (simp add: integral_completion)
moreover have "(f' has_integral integral\<^sup>L lborel f') UNIV \<longleftrightarrow> (f has_integral integral\<^sup>L lborel f') UNIV"
using f' by (intro has_integral_AE) auto
ultimately show ?thesis
by auto
qed
lemma has_integral_integral_lebesgue:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "integrable lebesgue f"
shows "(f has_integral (integral\<^sup>L lebesgue 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 lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b)) UNIV"
using f by (intro has_integral_sum finite_Basis ballI has_integral_scaleR_left has_integral_integral_lebesgue_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 lebesgue (\<lambda>x. f x \<bullet> b) *\<^sub>R b) = integral\<^sup>L lebesgue f"
using f by (subst (2) eq_f[symmetric]) simp
finally show ?thesis .
qed
lemma integrable_on_lebesgue:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "integrable lebesgue f \<Longrightarrow> f integrable_on UNIV"
using has_integral_integral_lebesgue by auto
lemma integral_lebesgue:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "integrable lebesgue f \<Longrightarrow> integral UNIV f = (\<integral>x. f x \<partial>lebesgue)"
using has_integral_integral_lebesgue by auto
end
subsection \<open>Absolute integrability (this is the same as Lebesgue integrability)\<close>
translations
"LBINT x. f" == "CONST lebesgue_integral CONST lborel (\<lambda>x. f)"
translations
"LBINT x:A. f" == "CONST set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
lemma set_integral_reflect:
fixes S and f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
shows "(LBINT x : S. f x) = (LBINT x : {x. - x \<in> S}. f (- x))"
by (subst lborel_integral_real_affine[where c="-1" and t=0])
(auto intro!: Bochner_Integration.integral_cong split: split_indicator)
lemma borel_integrable_atLeastAtMost':
fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
assumes f: "continuous_on {a..b} f"
shows "set_integrable lborel {a..b} f" (is "integrable _ ?f")
by (intro borel_integrable_compact compact_Icc f)
lemma integral_FTC_atLeastAtMost:
fixes f :: "real \<Rightarrow> 'a :: euclidean_space"
assumes "a \<le> b"
and F: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (F has_vector_derivative f x) (at x within {a .. b})"
and f: "continuous_on {a .. b} f"
shows "integral\<^sup>L lborel (\<lambda>x. indicator {a .. b} x *\<^sub>R f x) = F b - F a"
proof -
let ?f = "\<lambda>x. indicator {a .. b} x *\<^sub>R f x"
have "(?f has_integral (\<integral>x. ?f x \<partial>lborel)) UNIV"
using borel_integrable_atLeastAtMost'[OF f] by (rule has_integral_integral_lborel)
moreover
have "(f has_integral F b - F a) {a .. b}"
by (intro fundamental_theorem_of_calculus ballI assms) auto
then have "(?f has_integral F b - F a) {a .. b}"
by (subst has_integral_cong[where g=f]) auto
then have "(?f has_integral F b - F a) UNIV"
by (intro has_integral_on_superset[where T=UNIV and S="{a..b}"]) auto
ultimately show "integral\<^sup>L lborel ?f = F b - F a"
by (rule has_integral_unique)
qed
lemma set_borel_integral_eq_integral:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "set_integrable lborel S f"
shows "f integrable_on S" "LINT x : S | lborel. f x = integral S f"
proof -
let ?f = "\<lambda>x. indicator S x *\<^sub>R f x"
have "(?f has_integral LINT x : S | lborel. f x) UNIV"
by (rule has_integral_integral_lborel) fact
hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
apply (subst has_integral_restrict_UNIV [symmetric])
apply (rule has_integral_eq)
by auto
thus "f integrable_on S"
by (auto simp add: integrable_on_def)
with 1 have "(f has_integral (integral S f)) S"
by (intro integrable_integral, auto simp add: integrable_on_def)
thus "LINT x : S | lborel. f x = integral S f"
by (intro has_integral_unique [OF 1])
qed
lemma has_integral_set_lebesgue:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "set_integrable lebesgue S f"
shows "(f has_integral (LINT x:S|lebesgue. f x)) S"
using has_integral_integral_lebesgue[OF f]
by (simp_all add: indicator_def if_distrib[of "\<lambda>x. x *\<^sub>R f _"] has_integral_restrict_UNIV cong: if_cong)
lemma set_lebesgue_integral_eq_integral:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "set_integrable lebesgue S f"
shows "f integrable_on S" "LINT x:S | lebesgue. f x = integral S f"
using has_integral_set_lebesgue[OF f] by (auto simp: integral_unique integrable_on_def)
lemma lmeasurable_iff_has_integral:
"S \<in> lmeasurable \<longleftrightarrow> ((indicator S) has_integral measure lebesgue S) UNIV"
by (subst has_integral_iff_nn_integral_lebesgue)
(auto simp: ennreal_indicator emeasure_eq_measure2 borel_measurable_indicator_iff intro!: fmeasurableI)
abbreviation
absolutely_integrable_on :: "('a::euclidean_space \<Rightarrow> 'b::{banach, second_countable_topology}) \<Rightarrow> 'a set \<Rightarrow> bool"
(infixr "absolutely'_integrable'_on" 46)
where "f absolutely_integrable_on s \<equiv> set_integrable lebesgue s f"
lemma absolutely_integrable_on_def:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. norm (f x)) integrable_on s"
proof safe
assume f: "f absolutely_integrable_on s"
then have nf: "integrable lebesgue (\<lambda>x. norm (indicator s x *\<^sub>R f x))"
by (intro integrable_norm)
note integrable_on_lebesgue[OF f] integrable_on_lebesgue[OF nf]
moreover have
"(\<lambda>x. indicator s x *\<^sub>R f x) = (\<lambda>x. if x \<in> s then f x else 0)"
"(\<lambda>x. norm (indicator s x *\<^sub>R f x)) = (\<lambda>x. if x \<in> s then norm (f x) else 0)"
by auto
ultimately show "f integrable_on s" "(\<lambda>x. norm (f x)) integrable_on s"
by (simp_all add: integrable_restrict_UNIV)
next
assume f: "f integrable_on s" and nf: "(\<lambda>x. norm (f x)) integrable_on s"
show "f absolutely_integrable_on s"
proof (rule integrableI_bounded)
show "(\<lambda>x. indicator s x *\<^sub>R f x) \<in> borel_measurable lebesgue"
using f has_integral_implies_lebesgue_measurable[of f _ s] by (auto simp: integrable_on_def)
show "(\<integral>\<^sup>+ x. ennreal (norm (indicator s x *\<^sub>R f x)) \<partial>lebesgue) < \<infinity>"
using nf nn_integral_has_integral_lebesgue[of "\<lambda>x. norm (f x)" _ s]
by (auto simp: integrable_on_def nn_integral_completion)
qed
qed
lemma absolutely_integrable_on_null [intro]:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "content (cbox a b) = 0 \<Longrightarrow> f absolutely_integrable_on (cbox a b)"
by (auto simp: absolutely_integrable_on_def)
lemma absolutely_integrable_on_open_interval:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
shows "f absolutely_integrable_on box a b \<longleftrightarrow>
f absolutely_integrable_on cbox a b"
by (auto simp: integrable_on_open_interval absolutely_integrable_on_def)
lemma absolutely_integrable_restrict_UNIV:
"(\<lambda>x. if x \<in> s then f x else 0) absolutely_integrable_on UNIV \<longleftrightarrow> f absolutely_integrable_on s"
by (intro arg_cong2[where f=integrable]) auto
lemma absolutely_integrable_onI:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f integrable_on s \<Longrightarrow> (\<lambda>x. norm (f x)) integrable_on s \<Longrightarrow> f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def by auto
lemma nonnegative_absolutely_integrable_1:
fixes f :: "'a :: euclidean_space \<Rightarrow> real"
assumes f: "f integrable_on A" and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x"
shows "f absolutely_integrable_on A"
apply (rule absolutely_integrable_onI [OF f])
using assms by (simp add: integrable_eq)
lemma absolutely_integrable_on_iff_nonneg:
fixes f :: "'a :: euclidean_space \<Rightarrow> real"
assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "f absolutely_integrable_on S \<longleftrightarrow> f integrable_on S"
proof -
{ assume "f integrable_on S"
then have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on UNIV"
by (simp add: integrable_restrict_UNIV)
then have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV"
using \<open>f integrable_on S\<close> absolutely_integrable_restrict_UNIV assms nonnegative_absolutely_integrable_1 by blast
then have "f absolutely_integrable_on S"
using absolutely_integrable_restrict_UNIV by blast
}
then show ?thesis
unfolding absolutely_integrable_on_def by auto
qed
lemma lmeasurable_iff_integrable_on: "S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. 1::real) integrable_on S"
by (subst absolutely_integrable_on_iff_nonneg[symmetric])
(simp_all add: lmeasurable_iff_integrable)
lemma lmeasure_integral_UNIV: "S \<in> lmeasurable \<Longrightarrow> measure lebesgue S = integral UNIV (indicator S)"
by (simp add: lmeasurable_iff_has_integral integral_unique)
lemma lmeasure_integral: "S \<in> lmeasurable \<Longrightarrow> measure lebesgue S = integral S (\<lambda>x. 1::real)"
by (auto simp add: lmeasure_integral_UNIV indicator_def[abs_def] lmeasurable_iff_integrable_on)
lemma
assumes \<D>: "\<D> division_of S"
shows lmeasurable_division: "S \<in> lmeasurable" (is ?l)
and content_division: "(\<Sum>k\<in>\<D>. measure lebesgue k) = measure lebesgue S" (is ?m)
proof -
{ fix d1 d2 assume *: "d1 \<in> \<D>" "d2 \<in> \<D>" "d1 \<noteq> d2"
then obtain a b c d where "d1 = cbox a b" "d2 = cbox c d"
using division_ofD(4)[OF \<D>] by blast
with division_ofD(5)[OF \<D> *]
have "d1 \<in> sets lborel" "d2 \<in> sets lborel" "d1 \<inter> d2 \<subseteq> (cbox a b - box a b) \<union> (cbox c d - box c d)"
by auto
moreover have "(cbox a b - box a b) \<union> (cbox c d - box c d) \<in> null_sets lborel"
by (intro null_sets.Un null_sets_cbox_Diff_box)
ultimately have "d1 \<inter> d2 \<in> null_sets lborel"
by (blast intro: null_sets_subset) }
then show ?l ?m
unfolding division_ofD(6)[OF \<D>, symmetric]
using division_ofD(1,4)[OF \<D>]
by (auto intro!: measure_Union_AE[symmetric] simp: completion.AE_iff_null_sets Int_def[symmetric] pairwise_def null_sets_def)
qed
text \<open>This should be an abbreviation for negligible.\<close>
lemma negligible_iff_null_sets: "negligible S \<longleftrightarrow> S \<in> null_sets lebesgue"
proof
assume "negligible S"
then have "(indicator S has_integral (0::real)) UNIV"
by (auto simp: negligible)
then show "S \<in> null_sets lebesgue"
by (subst (asm) has_integral_iff_nn_integral_lebesgue)
(auto simp: borel_measurable_indicator_iff nn_integral_0_iff_AE AE_iff_null_sets indicator_eq_0_iff)
next
assume S: "S \<in> null_sets lebesgue"
show "negligible S"
unfolding negligible_def
proof (safe intro!: has_integral_iff_nn_integral_lebesgue[THEN iffD2]
has_integral_restrict_UNIV[where s="cbox _ _", THEN iffD1])
fix a b
show "(\<lambda>x. if x \<in> cbox a b then indicator S x else 0) \<in> lebesgue \<rightarrow>\<^sub>M borel"
using S by (auto intro!: measurable_If)
then show "(\<integral>\<^sup>+ x. ennreal (if x \<in> cbox a b then indicator S x else 0) \<partial>lebesgue) = ennreal 0"
using S[THEN AE_not_in] by (auto intro!: nn_integral_0_iff_AE[THEN iffD2])
qed auto
qed
lemma starlike_negligible:
assumes "closed S"
and eq1: "\<And>c x. \<lbrakk>(a + c *\<^sub>R x) \<in> S; 0 \<le> c; a + x \<in> S\<rbrakk> \<Longrightarrow> c = 1"
shows "negligible S"
proof -
have "negligible (op + (-a) ` S)"
proof (subst negligible_on_intervals, intro allI)
fix u v
show "negligible (op + (- a) ` S \<inter> cbox u v)"
unfolding negligible_iff_null_sets
apply (rule starlike_negligible_compact)
apply (simp add: assms closed_translation closed_Int_compact, clarify)
by (metis eq1 minus_add_cancel)
qed
then show ?thesis
by (rule negligible_translation_rev)
qed
lemma starlike_negligible_strong:
assumes "closed S"
and star: "\<And>c x. \<lbrakk>0 \<le> c; c < 1; a+x \<in> S\<rbrakk> \<Longrightarrow> a + c *\<^sub>R x \<notin> S"
shows "negligible S"
proof -
show ?thesis
proof (rule starlike_negligible [OF \<open>closed S\<close>, of a])
fix c x
assume cx: "a + c *\<^sub>R x \<in> S" "0 \<le> c" "a + x \<in> S"
with star have "~ (c < 1)" by auto
moreover have "~ (c > 1)"
using star [of "1/c" "c *\<^sub>R x"] cx by force
ultimately show "c = 1" by arith
qed
qed
subsection\<open>Applications\<close>
lemma negligible_hyperplane:
assumes "a \<noteq> 0 \<or> b \<noteq> 0" shows "negligible {x. a \<bullet> x = b}"
proof -
obtain x where x: "a \<bullet> x \<noteq> b"
using assms
apply auto
apply (metis inner_eq_zero_iff inner_zero_right)
using inner_zero_right by fastforce
show ?thesis
apply (rule starlike_negligible [OF closed_hyperplane, of x])
using x apply (auto simp: algebra_simps)
done
qed
lemma negligible_lowdim:
fixes S :: "'N :: euclidean_space set"
assumes "dim S < DIM('N)"
shows "negligible S"
proof -
obtain a where "a \<noteq> 0" and a: "span S \<subseteq> {x. a \<bullet> x = 0}"
using lowdim_subset_hyperplane [OF assms] by blast
have "negligible (span S)"
using \<open>a \<noteq> 0\<close> a negligible_hyperplane by (blast intro: negligible_subset)
then show ?thesis
using span_inc by (blast intro: negligible_subset)
qed
proposition negligible_convex_frontier:
fixes S :: "'N :: euclidean_space set"
assumes "convex S"
shows "negligible(frontier S)"
proof -
have nf: "negligible(frontier S)" if "convex S" "0 \<in> S" for S :: "'N set"
proof -
obtain B where "B \<subseteq> S" and indB: "independent B"
and spanB: "S \<subseteq> span B" and cardB: "card B = dim S"
by (metis basis_exists)
consider "dim S < DIM('N)" | "dim S = DIM('N)"
using dim_subset_UNIV le_eq_less_or_eq by blast
then show ?thesis
proof cases
case 1
show ?thesis
by (rule negligible_subset [of "closure S"])
(simp_all add: Diff_subset frontier_def negligible_lowdim 1)
next
case 2
obtain a where a: "a \<in> interior S"
apply (rule interior_simplex_nonempty [OF indB])
apply (simp add: indB independent_finite)
apply (simp add: cardB 2)
apply (metis \<open>B \<subseteq> S\<close> \<open>0 \<in> S\<close> \<open>convex S\<close> insert_absorb insert_subset interior_mono subset_hull)
done
show ?thesis
proof (rule starlike_negligible_strong [where a=a])
fix c::real and x
have eq: "a + c *\<^sub>R x = (a + x) - (1 - c) *\<^sub>R ((a + x) - a)"
by (simp add: algebra_simps)
assume "0 \<le> c" "c < 1" "a + x \<in> frontier S"
then show "a + c *\<^sub>R x \<notin> frontier S"
apply (clarsimp simp: frontier_def)
apply (subst eq)
apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a, of _ "1-c"], auto)
done
qed auto
qed
qed
show ?thesis
proof (cases "S = {}")
case True then show ?thesis by auto
next
case False
then obtain a where "a \<in> S" by auto
show ?thesis
using nf [of "(\<lambda>x. -a + x) ` S"]
by (metis \<open>a \<in> S\<close> add.left_inverse assms convex_translation_eq frontier_translation
image_eqI negligible_translation_rev)
qed
qed
corollary negligible_sphere: "negligible (sphere a e)"
using frontier_cball negligible_convex_frontier convex_cball
by (blast intro: negligible_subset)
lemma non_negligible_UNIV [simp]: "\<not> negligible UNIV"
unfolding negligible_iff_null_sets by (auto simp: null_sets_def emeasure_lborel_UNIV)
lemma negligible_interval:
"negligible (cbox a b) \<longleftrightarrow> box a b = {}" "negligible (box a b) \<longleftrightarrow> box a b = {}"
by (auto simp: negligible_iff_null_sets null_sets_def prod_nonneg inner_diff_left box_eq_empty
not_le emeasure_lborel_cbox_eq emeasure_lborel_box_eq
intro: eq_refl antisym less_imp_le)
subsection \<open>Negligibility of a Lipschitz image of a negligible set\<close>
lemma measure_eq_0_null_sets: "S \<in> null_sets M \<Longrightarrow> measure M S = 0"
by (auto simp: measure_def null_sets_def)
text\<open>The bound will be eliminated by a sort of onion argument\<close>
lemma locally_Lipschitz_negl_bounded:
fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
assumes MleN: "DIM('M) \<le> DIM('N)" "0 < B" "bounded S" "negligible S"
and lips: "\<And>x. x \<in> S
\<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and>
(\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
shows "negligible (f ` S)"
unfolding negligible_iff_null_sets
proof (clarsimp simp: completion.null_sets_outer)
fix e::real
assume "0 < e"
have "S \<in> lmeasurable"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
have e22: "0 < e / 2 / (2 * B * real DIM('M)) ^ DIM('N)"
using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: divide_simps)
obtain T
where "open T" "S \<subseteq> T" "T \<in> lmeasurable"
and "measure lebesgue T \<le> measure lebesgue S + e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
by (rule lmeasurable_outer_open [OF \<open>S \<in> lmeasurable\<close> e22])
then have T: "measure lebesgue T \<le> e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets measure_eq_0_null_sets)
have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
(x \<in> S \<longrightarrow> (\<forall>y. norm(y - x) < r
\<longrightarrow> y \<in> T \<and> (y \<in> S \<longrightarrow> norm(f y - f x) \<le> B * norm(y - x))))"
for x
proof (cases "x \<in> S")
case True
obtain U where "open U" "x \<in> U" and U: "\<And>y. y \<in> S \<inter> U \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
using lips [OF \<open>x \<in> S\<close>] by auto
have "x \<in> T \<inter> U"
using \<open>S \<subseteq> T\<close> \<open>x \<in> U\<close> \<open>x \<in> S\<close> by auto
then obtain \<epsilon> where "0 < \<epsilon>" "ball x \<epsilon> \<subseteq> T \<inter> U"
by (metis \<open>open T\<close> \<open>open U\<close> openE open_Int)
then show ?thesis
apply (rule_tac x="min (1/2) \<epsilon>" in exI)
apply (simp del: divide_const_simps)
apply (intro allI impI conjI)
apply (metis dist_commute dist_norm mem_ball subsetCE)
by (metis Int_iff subsetCE U dist_norm mem_ball norm_minus_commute)
next
case False
then show ?thesis
by (rule_tac x="1/4" in exI) auto
qed
then obtain R where R12: "\<And>x. 0 < R x \<and> R x \<le> 1/2"
and RT: "\<And>x y. \<lbrakk>x \<in> S; norm(y - x) < R x\<rbrakk> \<Longrightarrow> y \<in> T"
and RB: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; norm(y - x) < R x\<rbrakk> \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
by metis+
then have gaugeR: "gauge (\<lambda>x. ball x (R x))"
by (simp add: gauge_def)
obtain c where c: "S \<subseteq> cbox (-c *\<^sub>R One) (c *\<^sub>R One)" "box (-c *\<^sub>R One:: 'M) (c *\<^sub>R One) \<noteq> {}"
proof -
obtain B where B: "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
using \<open>bounded S\<close> bounded_iff by blast
show ?thesis
apply (rule_tac c = "abs B + 1" in that)
using norm_bound_Basis_le Basis_le_norm
apply (fastforce simp: box_eq_empty mem_box dest!: B intro: order_trans)+
done
qed
obtain \<D> where "countable \<D>"
and Dsub: "\<Union>\<D> \<subseteq> cbox (-c *\<^sub>R One) (c *\<^sub>R One)"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
and pw: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
and Ksub: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> (\<lambda>x. ball x (R x)) x"
and exN: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (2*c) / 2^n"
and "S \<subseteq> \<Union>\<D>"
using covering_lemma [OF c gaugeR] by force
have "\<exists>u v z. K = cbox u v \<and> box u v \<noteq> {} \<and> z \<in> S \<and> z \<in> cbox u v \<and>
cbox u v \<subseteq> ball z (R z)" if "K \<in> \<D>" for K
proof -
obtain u v where "K = cbox u v"
using \<open>K \<in> \<D>\<close> cbox by blast
with that show ?thesis
apply (rule_tac x=u in exI)
apply (rule_tac x=v in exI)
apply (metis Int_iff interior_cbox cbox Ksub)
done
qed
then obtain uf vf zf
where uvz: "\<And>K. K \<in> \<D> \<Longrightarrow>
K = cbox (uf K) (vf K) \<and> box (uf K) (vf K) \<noteq> {} \<and> zf K \<in> S \<and>
zf K \<in> cbox (uf K) (vf K) \<and> cbox (uf K) (vf K) \<subseteq> ball (zf K) (R (zf K))"
by metis
define prj1 where "prj1 \<equiv> \<lambda>x::'M. x \<bullet> (SOME i. i \<in> Basis)"
define fbx where "fbx \<equiv> \<lambda>D. cbox (f(zf D) - (B * DIM('M) * (prj1(vf D - uf D))) *\<^sub>R One::'N)
(f(zf D) + (B * DIM('M) * prj1(vf D - uf D)) *\<^sub>R One)"
have vu_pos: "0 < prj1 (vf X - uf X)" if "X \<in> \<D>" for X
using uvz [OF that] by (simp add: prj1_def box_ne_empty SOME_Basis inner_diff_left)
have prj1_idem: "prj1 (vf X - uf X) = (vf X - uf X) \<bullet> i" if "X \<in> \<D>" "i \<in> Basis" for X i
proof -
have "cbox (uf X) (vf X) \<in> \<D>"
using uvz \<open>X \<in> \<D>\<close> by auto
with exN obtain n where "\<And>i. i \<in> Basis \<Longrightarrow> vf X \<bullet> i - uf X \<bullet> i = (2*c) / 2^n"
by blast
then show ?thesis
by (simp add: \<open>i \<in> Basis\<close> SOME_Basis inner_diff prj1_def)
qed
have countbl: "countable (fbx ` \<D>)"
using \<open>countable \<D>\<close> by blast
have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e / 2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
proof -
have BM_ge0: "0 \<le> B * (DIM('M) * prj1 (vf X - uf X))" if "X \<in> \<D>'" for X
using \<open>0 < B\<close> \<open>\<D>' \<subseteq> \<D>\<close> that vu_pos by fastforce
have "{} \<notin> \<D>'"
using cbox \<open>\<D>' \<subseteq> \<D>\<close> interior_empty by blast
have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> sum (measure lebesgue o fbx) \<D>'"
by (rule sum_image_le [OF \<open>finite \<D>'\<close>]) (force simp: fbx_def)
also have "\<dots> \<le> (\<Sum>X\<in>\<D>'. (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X)"
proof (rule sum_mono)
fix X assume "X \<in> \<D>'"
then have "X \<in> \<D>" using \<open>\<D>' \<subseteq> \<D>\<close> by blast
then have ufvf: "cbox (uf X) (vf X) = X"
using uvz by blast
have "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i::'M \<in> Basis. prj1 (vf X - uf X))"
by (rule prod_constant [symmetric])
also have "\<dots> = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)"
using prj1_idem [OF \<open>X \<in> \<D>\<close>] by (auto simp: algebra_simps intro: prod.cong)
finally have prj1_eq: "prj1 (vf X - uf X) ^ DIM('M) = (\<Prod>i\<in>Basis. vf X \<bullet> i - uf X \<bullet> i)" .
have "uf X \<in> cbox (uf X) (vf X)" "vf X \<in> cbox (uf X) (vf X)"
using uvz [OF \<open>X \<in> \<D>\<close>] by (force simp: mem_box)+
moreover have "cbox (uf X) (vf X) \<subseteq> ball (zf X) (1/2)"
by (meson R12 order_trans subset_ball uvz [OF \<open>X \<in> \<D>\<close>])
ultimately have "uf X \<in> ball (zf X) (1/2)" "vf X \<in> ball (zf X) (1/2)"
by auto
then have "dist (vf X) (uf X) \<le> 1"
unfolding mem_ball
by (metis dist_commute dist_triangle_half_l dual_order.order_iff_strict)
then have 1: "prj1 (vf X - uf X) \<le> 1"
unfolding prj1_def dist_norm using Basis_le_norm SOME_Basis order_trans by fastforce
have 0: "0 \<le> prj1 (vf X - uf X)"
using \<open>X \<in> \<D>\<close> prj1_def vu_pos by fastforce
have "(measure lebesgue \<circ> fbx) X \<le> (2 * B * DIM('M)) ^ DIM('N) * content (cbox (uf X) (vf X))"
apply (simp add: fbx_def content_cbox_cases algebra_simps BM_ge0 \<open>X \<in> \<D>'\<close> prod_constant)
apply (simp add: power_mult_distrib \<open>0 < B\<close> prj1_eq [symmetric])
using MleN 0 1 uvz \<open>X \<in> \<D>\<close>
apply (fastforce simp add: box_ne_empty power_decreasing)
done
also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X"
by (subst (3) ufvf[symmetric]) simp
finally show "(measure lebesgue \<circ> fbx) X \<le> (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X" .
qed
also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>'"
by (simp add: sum_distrib_left)
also have "\<dots> \<le> e / 2"
proof -
have div: "\<D>' division_of \<Union>\<D>'"
apply (auto simp: \<open>finite \<D>'\<close> \<open>{} \<notin> \<D>'\<close> division_of_def)
using cbox that apply blast
using pairwise_subset [OF pw \<open>\<D>' \<subseteq> \<D>\<close>] unfolding pairwise_def apply force+
done
have le_meaT: "measure lebesgue (\<Union>\<D>') \<le> measure lebesgue T"
proof (rule measure_mono_fmeasurable [OF _ _ \<open>T : lmeasurable\<close>])
show "(\<Union>\<D>') \<in> sets lebesgue"
using div lmeasurable_division by auto
have "\<Union>\<D>' \<subseteq> \<Union>\<D>"
using \<open>\<D>' \<subseteq> \<D>\<close> by blast
also have "... \<subseteq> T"
proof (clarify)
fix x D
assume "x \<in> D" "D \<in> \<D>"
show "x \<in> T"
using Ksub [OF \<open>D \<in> \<D>\<close>]
by (metis \<open>x \<in> D\<close> Int_iff dist_norm mem_ball norm_minus_commute subsetD RT)
qed
finally show "\<Union>\<D>' \<subseteq> T" .
qed
have "sum (measure lebesgue) \<D>' = sum content \<D>'"
using \<open>\<D>' \<subseteq> \<D>\<close> cbox by (force intro: sum.cong)
then have "(2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>' =
(2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue (\<Union>\<D>')"
using content_division [OF div] by auto
also have "\<dots> \<le> (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue T"
apply (rule mult_left_mono [OF le_meaT])
using \<open>0 < B\<close>
apply (simp add: algebra_simps)
done
also have "\<dots> \<le> e / 2"
using T \<open>0 < B\<close> by (simp add: field_simps)
finally show ?thesis .
qed
finally show ?thesis .
qed
then have e2: "sum (measure lebesgue) \<G> \<le> e / 2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
by (metis finite_subset_image that)
show "\<exists>W\<in>lmeasurable. f ` S \<subseteq> W \<and> measure lebesgue W < e"
proof (intro bexI conjI)
have "\<exists>X\<in>\<D>. f y \<in> fbx X" if "y \<in> S" for y
proof -
obtain X where "y \<in> X" "X \<in> \<D>"
using \<open>S \<subseteq> \<Union>\<D>\<close> \<open>y \<in> S\<close> by auto
then have y: "y \<in> ball(zf X) (R(zf X))"
using uvz by fastforce
have conj_le_eq: "z - b \<le> y \<and> y \<le> z + b \<longleftrightarrow> abs(y - z) \<le> b" for z y b::real
by auto
have yin: "y \<in> cbox (uf X) (vf X)" and zin: "(zf X) \<in> cbox (uf X) (vf X)"
using uvz \<open>X \<in> \<D>\<close> \<open>y \<in> X\<close> by auto
have "norm (y - zf X) \<le> (\<Sum>i\<in>Basis. \<bar>(y - zf X) \<bullet> i\<bar>)"
by (rule norm_le_l1)
also have "\<dots> \<le> real DIM('M) * prj1 (vf X - uf X)"
proof (rule sum_bounded_above)
fix j::'M assume j: "j \<in> Basis"
show "\<bar>(y - zf X) \<bullet> j\<bar> \<le> prj1 (vf X - uf X)"
using yin zin j
by (fastforce simp add: mem_box prj1_idem [OF \<open>X \<in> \<D>\<close> j] inner_diff_left)
qed
finally have nole: "norm (y - zf X) \<le> DIM('M) * prj1 (vf X - uf X)"
by simp
have fle: "\<bar>f y \<bullet> i - f(zf X) \<bullet> i\<bar> \<le> B * DIM('M) * prj1 (vf X - uf X)" if "i \<in> Basis" for i
proof -
have "\<bar>f y \<bullet> i - f (zf X) \<bullet> i\<bar> = \<bar>(f y - f (zf X)) \<bullet> i\<bar>"
by (simp add: algebra_simps)
also have "\<dots> \<le> norm (f y - f (zf X))"
by (simp add: Basis_le_norm that)
also have "\<dots> \<le> B * norm(y - zf X)"
by (metis uvz RB \<open>X \<in> \<D>\<close> dist_commute dist_norm mem_ball \<open>y \<in> S\<close> y)
also have "\<dots> \<le> B * real DIM('M) * prj1 (vf X - uf X)"
using \<open>0 < B\<close> by (simp add: nole)
finally show ?thesis .
qed
show ?thesis
by (rule_tac x=X in bexI)
(auto simp: fbx_def prj1_idem mem_box conj_le_eq inner_add inner_diff fle \<open>X \<in> \<D>\<close>)
qed
then show "f ` S \<subseteq> (\<Union>D\<in>\<D>. fbx D)" by auto
next
have 1: "\<And>D. D \<in> \<D> \<Longrightarrow> fbx D \<in> lmeasurable"
by (auto simp: fbx_def)
have 2: "I' \<subseteq> \<D> \<Longrightarrow> finite I' \<Longrightarrow> measure lebesgue (\<Union>D\<in>I'. fbx D) \<le> e/2" for I'
by (rule order_trans[OF measure_Union_le e2]) (auto simp: fbx_def)
have 3: "0 \<le> e/2"
using \<open>0<e\<close> by auto
show "(\<Union>D\<in>\<D>. fbx D) \<in> lmeasurable"
by (intro fmeasurable_UN_bound[OF \<open>countable \<D>\<close> 1 2 3])
have "measure lebesgue (\<Union>D\<in>\<D>. fbx D) \<le> e/2"
by (intro measure_UN_bound[OF \<open>countable \<D>\<close> 1 2 3])
then show "measure lebesgue (\<Union>D\<in>\<D>. fbx D) < e"
using \<open>0 < e\<close> by linarith
qed
qed
proposition negligible_locally_Lipschitz_image:
fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
assumes MleN: "DIM('M) \<le> DIM('N)" "negligible S"
and lips: "\<And>x. x \<in> S
\<Longrightarrow> \<exists>T B. open T \<and> x \<in> T \<and>
(\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
shows "negligible (f ` S)"
proof -
let ?S = "\<lambda>n. ({x \<in> S. norm x \<le> n \<and>
(\<exists>T. open T \<and> x \<in> T \<and>
(\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> (real n + 1) * norm (y - x)))})"
have negfn: "f ` ?S n \<in> null_sets lebesgue" for n::nat
unfolding negligible_iff_null_sets[symmetric]
apply (rule_tac B = "real n + 1" in locally_Lipschitz_negl_bounded)
by (auto simp: MleN bounded_iff intro: negligible_subset [OF \<open>negligible S\<close>])
have "S = (\<Union>n. ?S n)"
proof (intro set_eqI iffI)
fix x assume "x \<in> S"
with lips obtain T B where T: "open T" "x \<in> T"
and B: "\<And>y. y \<in> S \<inter> T \<Longrightarrow> norm(f y - f x) \<le> B * norm(y - x)"
by metis+
have no: "norm (f y - f x) \<le> (nat \<lceil>max B (norm x)\<rceil> + 1) * norm (y - x)" if "y \<in> S \<inter> T" for y
proof -
have "B * norm(y - x) \<le> (nat \<lceil>max B (norm x)\<rceil> + 1) * norm (y - x)"
by (meson max.cobounded1 mult_right_mono nat_ceiling_le_eq nat_le_iff_add norm_ge_zero order_trans)
then show ?thesis
using B order_trans that by blast
qed
have "x \<in> ?S (nat (ceiling (max B (norm x))))"
apply (simp add: \<open>x \<in> S \<close>, rule)
using real_nat_ceiling_ge max.bounded_iff apply blast
using T no
apply (force simp: algebra_simps)
done
then show "x \<in> (\<Union>n. ?S n)" by force
qed auto
then show ?thesis
by (rule ssubst) (auto simp: image_Union negligible_iff_null_sets intro: negfn)
qed
corollary negligible_differentiable_image_negligible:
fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
assumes MleN: "DIM('M) \<le> DIM('N)" "negligible S"
and diff_f: "f differentiable_on S"
shows "negligible (f ` S)"
proof -
have "\<exists>T B. open T \<and> x \<in> T \<and> (\<forall>y \<in> S \<inter> T. norm(f y - f x) \<le> B * norm(y - x))"
if "x \<in> S" for x
proof -
obtain f' where "linear f'"
and f': "\<And>e. e>0 \<Longrightarrow>
\<exists>d>0. \<forall>y\<in>S. norm (y - x) < d \<longrightarrow>
norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)"
using diff_f \<open>x \<in> S\<close>
by (auto simp: linear_linear differentiable_on_def differentiable_def has_derivative_within_alt)
obtain B where "B > 0" and B: "\<forall>x. norm (f' x) \<le> B * norm x"
using linear_bounded_pos \<open>linear f'\<close> by blast
obtain d where "d>0"
and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - x) < d\<rbrakk> \<Longrightarrow>
norm (f y - f x - f' (y - x)) \<le> norm (y - x)"
using f' [of 1] by (force simp:)
have *: "norm (f y - f x) \<le> (B + 1) * norm (y - x)"
if "y \<in> S" "norm (y - x) < d" for y
proof -
have "norm (f y - f x) -B * norm (y - x) \<le> norm (f y - f x) - norm (f' (y - x))"
by (simp add: B)
also have "\<dots> \<le> norm (f y - f x - f' (y - x))"
by (rule norm_triangle_ineq2)
also have "... \<le> norm (y - x)"
by (rule d [OF that])
finally show ?thesis
by (simp add: algebra_simps)
qed
show ?thesis
apply (rule_tac x="ball x d" in exI)
apply (rule_tac x="B+1" in exI)
using \<open>d>0\<close>
apply (auto simp: dist_norm norm_minus_commute intro!: *)
done
qed
with negligible_locally_Lipschitz_image assms show ?thesis by metis
qed
corollary negligible_differentiable_image_lowdim:
fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
assumes MlessN: "DIM('M) < DIM('N)" and diff_f: "f differentiable_on S"
shows "negligible (f ` S)"
proof -
have "x \<le> DIM('M) \<Longrightarrow> x \<le> DIM('N)" for x
using MlessN by linarith
obtain lift :: "'M * real \<Rightarrow> 'N" and drop :: "'N \<Rightarrow> 'M * real" and j :: 'N
where "linear lift" "linear drop" and dropl [simp]: "\<And>z. drop (lift z) = z"
and "j \<in> Basis" and j: "\<And>x. lift(x,0) \<bullet> j = 0"
using lowerdim_embeddings [OF MlessN] by metis
have "negligible {x. x\<bullet>j = 0}"
by (metis \<open>j \<in> Basis\<close> negligible_standard_hyperplane)
then have neg0S: "negligible ((\<lambda>x. lift (x, 0)) ` S)"
apply (rule negligible_subset)
by (simp add: image_subsetI j)
have diff_f': "f \<circ> fst \<circ> drop differentiable_on (\<lambda>x. lift (x, 0)) ` S"
using diff_f
apply (clarsimp simp add: differentiable_on_def)
apply (intro differentiable_chain_within linear_imp_differentiable [OF \<open>linear drop\<close>]
linear_imp_differentiable [OF fst_linear])
apply (force simp: image_comp o_def)
done
have "f = (f o fst o drop o (\<lambda>x. lift (x, 0)))"
by (simp add: o_def)
then show ?thesis
apply (rule ssubst)
apply (subst image_comp [symmetric])
apply (metis negligible_differentiable_image_negligible order_refl diff_f' neg0S)
done
qed
lemma set_integral_norm_bound:
fixes f :: "_ \<Rightarrow> 'a :: {banach, second_countable_topology}"
shows "set_integrable M k f \<Longrightarrow> norm (LINT x:k|M. f x) \<le> LINT x:k|M. norm (f x)"
using integral_norm_bound[of M "\<lambda>x. indicator k x *\<^sub>R f x"] by simp
lemma set_integral_finite_UN_AE:
fixes f :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "finite I"
and ae: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> AE x in M. (x \<in> A i \<and> x \<in> A j) \<longrightarrow> i = j"
and [measurable]: "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M"
and f: "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f"
shows "LINT x:(\<Union>i\<in>I. A i)|M. f x = (\<Sum>i\<in>I. LINT x:A i|M. f x)"
using \<open>finite I\<close> order_refl[of I]
proof (induction I rule: finite_subset_induct')
case (insert i I')
have "AE x in M. (\<forall>j\<in>I'. x \<in> A i \<longrightarrow> x \<notin> A j)"
proof (intro AE_ball_countable[THEN iffD2] ballI)
fix j assume "j \<in> I'"
with \<open>I' \<subseteq> I\<close> \<open>i \<notin> I'\<close> have "i \<noteq> j" "j \<in> I"
by auto
then show "AE x in M. x \<in> A i \<longrightarrow> x \<notin> A j"
using ae[of i j] \<open>i \<in> I\<close> by auto
qed (use \<open>finite I'\<close> in \<open>rule countable_finite\<close>)
then have "AE x\<in>A i in M. \<forall>xa\<in>I'. x \<notin> A xa "
by auto
with insert.hyps insert.IH[symmetric]
show ?case
by (auto intro!: set_integral_Un_AE sets.finite_UN f set_integrable_UN)
qed simp
lemma set_integrable_norm:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
assumes f: "set_integrable M k f" shows "set_integrable M k (\<lambda>x. norm (f x))"
using integrable_norm[OF f] by simp
lemma absolutely_integrable_bounded_variation:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f absolutely_integrable_on UNIV"
obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
proof (rule that[of "integral UNIV (\<lambda>x. norm (f x))"]; safe)
fix d :: "'a set set" assume d: "d division_of \<Union>d"
have *: "k \<in> d \<Longrightarrow> f absolutely_integrable_on k" for k
using f[THEN set_integrable_subset, of k] division_ofD(2,4)[OF d, of k] by auto
note d' = division_ofD[OF d]
have "(\<Sum>k\<in>d. norm (integral k f)) = (\<Sum>k\<in>d. norm (LINT x:k|lebesgue. f x))"
by (intro sum.cong refl arg_cong[where f=norm] set_lebesgue_integral_eq_integral(2)[symmetric] *)
also have "\<dots> \<le> (\<Sum>k\<in>d. LINT x:k|lebesgue. norm (f x))"
by (intro sum_mono set_integral_norm_bound *)
also have "\<dots> = (\<Sum>k\<in>d. integral k (\<lambda>x. norm (f x)))"
by (intro sum.cong refl set_lebesgue_integral_eq_integral(2) set_integrable_norm *)
also have "\<dots> \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
using integrable_on_subdivision[OF d] assms f unfolding absolutely_integrable_on_def
by (subst integral_combine_division_topdown[OF _ d]) auto
also have "\<dots> \<le> integral UNIV (\<lambda>x. norm (f x))"
using integrable_on_subdivision[OF d] assms unfolding absolutely_integrable_on_def
by (intro integral_subset_le) auto
finally show "(\<Sum>k\<in>d. norm (integral k f)) \<le> integral UNIV (\<lambda>x. norm (f x))" .
qed
lemma helplemma:
assumes "sum (\<lambda>x. norm (f x - g x)) s < e"
and "finite s"
shows "\<bar>sum (\<lambda>x. norm(f x)) s - sum (\<lambda>x. norm(g x)) s\<bar> < e"
unfolding sum_subtractf[symmetric]
apply (rule le_less_trans[OF sum_abs])
apply (rule le_less_trans[OF _ assms(1)])
apply (rule sum_mono)
apply (rule norm_triangle_ineq3)
done
lemma bounded_variation_absolutely_integrable_interval:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
and *: "\<forall>d. d division_of (cbox a b) \<longrightarrow> sum (\<lambda>k. norm(integral k f)) d \<le> B"
shows "f absolutely_integrable_on cbox a b"
proof -
let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of (cbox a b)}"
have D_1: "?D \<noteq> {}"
by (rule elementary_interval[of a b]) auto
have D_2: "bdd_above (?f`?D)"
by (metis * mem_Collect_eq bdd_aboveI2)
note D = D_1 D_2
let ?S = "SUP x:?D. ?f x"
show ?thesis
apply (rule absolutely_integrable_onI [OF f has_integral_integrable])
apply (subst has_integral[of _ ?S])
apply safe
proof goal_cases
case e: (1 e)
then have "?S - e / 2 < ?S" by simp
then obtain d where d: "d division_of (cbox a b)" "?S - e / 2 < (\<Sum>k\<in>d. norm (integral k f))"
unfolding less_cSUP_iff[OF D] by auto
note d' = division_ofD[OF this(1)]
have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
proof
fix x
have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
apply (rule separate_point_closed)
apply (rule closed_Union)
apply (rule finite_subset[OF _ d'(1)])
using d'(4)
apply auto
done
then show "\<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
by force
qed
from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
have "e/2 > 0"
using e by auto
from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
show ?case
apply (rule_tac x="?g" in exI)
apply safe
proof -
show "gauge ?g"
using g(1) k(1)
unfolding gauge_def
by auto
fix p
assume "p tagged_division_of (cbox a b)" and "?g fine p"
note p = this(1) conjunctD2[OF this(2)[unfolded fine_Int]]
note p' = tagged_division_ofD[OF p(1)]
define p' where "p' = {(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
have gp': "g fine p'"
using p(2)
unfolding p'_def fine_def
by auto
have p'': "p' tagged_division_of (cbox a b)"
apply (rule tagged_division_ofI)
proof -
show "finite p'"
apply (rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l)) `
{(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"])
unfolding p'_def
defer
apply (rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
apply safe
unfolding image_iff
apply (rule_tac x="(i,x,l)" in bexI)
apply auto
done
fix x k
assume "(x, k) \<in> p'"
then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
unfolding p'_def by auto
then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
show "x \<in> k" and "k \<subseteq> cbox a b"
using p'(2-3)[OF il(3)] il by auto
show "\<exists>a b. k = cbox a b"
unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
apply safe
unfolding Int_interval
apply auto
done
next
fix x1 k1
assume "(x1, k1) \<in> p'"
then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> k1 = i \<inter> l"
unfolding p'_def by auto
then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "k1 = i1 \<inter> l1" by blast
fix x2 k2
assume "(x2,k2)\<in>p'"
then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> k2 = i \<inter> l"
unfolding p'_def by auto
then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "k2 = i2 \<inter> l2" by blast
assume "(x1, k1) \<noteq> (x2, k2)"
then have "interior i1 \<inter> interior i2 = {} \<or> interior l1 \<inter> interior l2 = {}"
using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)]
unfolding il1 il2
by auto
then show "interior k1 \<inter> interior k2 = {}"
unfolding il1 il2 by auto
next
have *: "\<forall>(x, X) \<in> p'. X \<subseteq> cbox a b"
unfolding p'_def using d' by auto
show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = cbox a b"
apply rule
apply (rule Union_least)
unfolding mem_Collect_eq
apply (erule exE)
apply (drule *[rule_format])
apply safe
proof -
fix y
assume y: "y \<in> cbox a b"
then have "\<exists>x l. (x, l) \<in> p \<and> y\<in>l"
unfolding p'(6)[symmetric] by auto
then guess x l by (elim exE) note xl=conjunctD2[OF this]
then have "\<exists>k. k \<in> d \<and> y \<in> k"
using y unfolding d'(6)[symmetric] by auto
then guess i .. note i = conjunctD2[OF this]
have "x \<in> i"
using fineD[OF p(3) xl(1)]
using k(2)[OF i(1), of x]
using i(2) xl(2)
by auto
then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
unfolding p'_def Union_iff
apply (rule_tac x="i \<inter> l" in bexI)
using i xl
apply auto
done
qed
qed
then have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
apply -
apply (rule g(2)[rule_format])
unfolding tagged_division_of_def
apply safe
apply (rule gp')
done
then have **: "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
unfolding split_def
using p''
by (force intro!: helplemma)
have p'alt: "p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}}"
proof (safe, goal_cases)
case prems: (2 _ _ x i l)
have "x \<in> i"
using fineD[OF p(3) prems(1)] k(2)[OF prems(2), of x] prems(4-)
by auto
then have "(x, i \<inter> l) \<in> p'"
unfolding p'_def
using prems
apply safe
apply (rule_tac x=x in exI)
apply (rule_tac x="i \<inter> l" in exI)
apply safe
using prems
apply auto
done
then show ?case
using prems(3) by auto
next
fix x k
assume "(x, k) \<in> p'"
then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
unfolding p'_def by auto
then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
then show "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
using p'(2) by fastforce
qed
have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
apply (auto intro: integral_null simp: content_eq_0_interior)
done
note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
have *: "\<And>sni sni' sf sf'. \<bar>sf' - sni'\<bar> < e / 2 \<longrightarrow> ?S - e / 2 < sni \<and> sni' \<le> ?S \<and>
sni \<le> sni' \<and> sf' = sf \<longrightarrow> \<bar>sf - ?S\<bar> < e"
by arith
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e"
unfolding real_norm_def
apply (rule *[rule_format,OF **])
apply safe
apply(rule d(2))
proof goal_cases
case 1
show ?case
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
next
case 2
have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
(\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}"
by auto
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
proof (rule sum_mono, goal_cases)
case k: (1 k)
from d'(4)[OF this] guess u v by (elim exE) note uv=this
define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}"
note uvab = d'(2)[OF k[unfolded uv]]
have "d' division_of cbox u v"
apply (subst d'_def)
apply (rule division_inter_1)
apply (rule division_of_tagged_division[OF p(1)])
apply (rule uvab)
done
then have "norm (integral k f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
unfolding uv
apply (subst integral_combine_division_topdown[of _ _ d'])
apply (rule integrable_on_subcbox[OF assms(1) uvab])
apply assumption
apply (rule sum_norm_le)
apply auto
done
also have "\<dots> = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))"
apply (rule sum.mono_neutral_left)
apply (subst Setcompr_eq_image)
apply (rule finite_imageI)+
apply fact
unfolding d'_def uv
apply blast
proof (rule, goal_cases)
case prems: (1 i)
then have "i \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
by auto
from this[unfolded mem_Collect_eq] guess l .. note l=this
then have "cbox u v \<inter> l = {}"
using prems by auto
then show ?case
using l by auto
qed
also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))"
unfolding Setcompr_eq_image
apply (rule sum.reindex_nontrivial [unfolded o_def])
apply (rule finite_imageI)
apply (rule p')
proof goal_cases
case prems: (1 l y)
have "interior (k \<inter> l) \<subseteq> interior (l \<inter> y)"
apply (subst(2) interior_Int)
by (metis Int_lower2 Int_subset_iff interior_mono prems(4))
then have *: "interior (k \<inter> l) = {}"
using snd_p(5)[OF prems(1-3)] by auto
from d'(4)[OF k] snd_p(4)[OF prems(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
show ?case
using *
unfolding uv Int_interval content_eq_0_interior[symmetric]
by auto
qed
finally show ?case .
qed
also have "\<dots> = (\<Sum>(i,l)\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (i\<inter>l) f))"
apply (subst sum_sum_product[symmetric])
apply fact
using p'(1)
apply auto
done
also have "\<dots> = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (case_prod op \<inter> x) f))"
unfolding split_def ..
also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
unfolding *
apply (rule sum.reindex_nontrivial [symmetric, unfolded o_def])
apply (rule finite_product_dependent)
apply fact
apply (rule finite_imageI)
apply (rule p')
unfolding split_paired_all mem_Collect_eq split_conv o_def
proof -
note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)]
fix l1 l2 k1 k2
assume as:
"(l1, k1) \<noteq> (l2, k2)"
"l1 \<inter> k1 = l2 \<inter> k2"
"\<exists>i l. (l1, k1) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
"\<exists>i l. (l2, k2) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
then have "l1 \<in> d" and "k1 \<in> snd ` p"
by auto from d'(4)[OF this(1)] *(1)[OF this(2)]
guess u1 v1 u2 v2 by (elim exE) note uv=this
have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
using as by auto
then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
by (metis Pair_inject \<open>k1 \<in> snd ` p\<close> \<open>l1 \<in> d\<close> as(4) d'(5) snd_p(5))
moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
using as(2) by auto
ultimately have "interior(l1 \<inter> k1) = {}"
by auto
then show "norm (integral (l1 \<inter> k1) f) = 0"
unfolding uv Int_interval
unfolding content_eq_0_interior[symmetric]
by auto
qed
also have "\<dots> = (\<Sum>(x, k)\<in>p'. norm (integral k f))"
unfolding sum_p'
apply (rule sum.mono_neutral_right)
apply (subst *)
apply (rule finite_imageI[OF finite_product_dependent])
apply fact
apply (rule finite_imageI[OF p'(1)])
apply safe
proof goal_cases
case (2 i ia l a b)
then have "ia \<inter> b = {}"
unfolding p'alt image_iff Bex_def not_ex
apply (erule_tac x="(a, ia \<inter> b)" in allE)
apply auto
done
then show ?case
by auto
next
case (1 x a b)
then show ?case
unfolding p'_def
proof -
assume "(a, b) \<in> {(x, k) |x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l}"
then have "\<exists>n N. (a, b) = (n, N) \<and> (\<exists>Na Nb. n \<in> Na \<and> Na \<in> d \<and> (n, Nb) \<in> p \<and> N = Na \<inter> Nb)"
by force
then show ?thesis
by (metis (no_types) image_iff snd_conv)
qed
qed
finally show ?case .
next
case 3
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
have Sigma_alt: "\<And>s t. s \<times> t = {(i, j) |i j. i \<in> s \<and> j \<in> t}"
by auto
have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
apply safe
unfolding image_iff
apply (rule_tac x="((x,l),i)" in bexI)
apply auto
done
note pdfin = finite_cartesian_product[OF p'(1) d'(1)]
have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
unfolding norm_scaleR
apply (rule sum.mono_neutral_left)
apply (subst *)
apply (rule finite_imageI)
apply fact
unfolding p'alt
apply blast
apply safe
apply (rule_tac x=x in exI)
apply (rule_tac x=i in exI)
apply (rule_tac x=l in exI)
apply auto
done
also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
unfolding *
apply (subst sum.reindex_nontrivial)
apply fact
unfolding split_paired_all
unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff prod.inject
apply (elim conjE)
proof -
fix x1 l1 k1 x2 l2 k2
assume as: "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
"x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> k1 = k2)"
from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
by auto
then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
apply -
apply (erule disjE)
apply (rule disjI2)
defer
apply (rule disjI1)
apply (rule d'(5)[OF as(3-4)])
apply assumption
apply (rule p'(5)[OF as(1-2)])
apply auto
done
moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
unfolding as ..
ultimately have "interior (l1 \<inter> k1) = {}"
by auto
then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
unfolding uv Int_interval
unfolding content_eq_0_interior[symmetric]
by auto
qed safe
also have "\<dots> = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
unfolding Sigma_alt
apply (subst sum_sum_product[symmetric])
apply (rule p')
apply rule
apply (rule d')
apply (rule sum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
proof -
fix x l
assume as: "(x, l) \<in> p"
note xl = p'(2-4)[OF this]
from this(3) guess u v by (elim exE) note uv=this
have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
by (simp add: Int_commute uv)
also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
unfolding Setcompr_eq_image
apply (rule sum.reindex_nontrivial [unfolded o_def, symmetric])
apply (rule d')
proof goal_cases
case prems: (1 k y)
from d'(4)[OF this(1)] d'(4)[OF this(2)]
guess u1 v1 u2 v2 by (elim exE) note uv=this
have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
apply (subst interior_Int)
using d'(5)[OF prems(1-3)]
apply auto
done
also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
by auto
also have "\<dots> = interior (k \<inter> cbox u v)"
unfolding prems(4) by auto
finally show ?case
unfolding uv Int_interval content_eq_0_interior ..
qed
also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
apply (rule sum.mono_neutral_right)
unfolding Setcompr_eq_image
apply (rule finite_imageI)
apply (rule d')
apply blast
apply safe
apply (rule_tac x=k in exI)
proof goal_cases
case prems: (1 i k)
from d'(4)[OF this(1)] guess a b by (elim exE) note ab=this
have "interior (k \<inter> cbox u v) \<noteq> {}"
using prems(2)
unfolding ab Int_interval content_eq_0_interior
by auto
then show ?case
using prems(1)
using interior_subset[of "k \<inter> cbox u v"]
by auto
qed
finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
unfolding sum_distrib_right[symmetric] real_scaleR_def
apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
using xl(2)[unfolded uv]
unfolding uv
apply auto
done
qed
finally show ?case .
qed
qed
qed
qed
lemma bounded_variation_absolutely_integrable:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "f integrable_on UNIV"
and "\<forall>d. d division_of (\<Union>d) \<longrightarrow> sum (\<lambda>k. norm (integral k f)) d \<le> B"
shows "f absolutely_integrable_on UNIV"
proof (rule absolutely_integrable_onI, fact)
let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of (\<Union>d)}"
have D_1: "?D \<noteq> {}"
by (rule elementary_interval) auto
have D_2: "bdd_above (?f`?D)"
by (intro bdd_aboveI2[where M=B] assms(2)[rule_format]) simp
note D = D_1 D_2
let ?S = "SUP d:?D. ?f d"
have "\<And>a b. f integrable_on cbox a b"
using assms(1) integrable_on_subcbox by blast
then have f_int: "\<And>a b. f absolutely_integrable_on cbox a b"
apply (rule bounded_variation_absolutely_integrable_interval[where B=B])
using assms(2) apply blast
done
have "((\<lambda>x. norm (f x)) has_integral ?S) UNIV"
apply (subst has_integral_alt')
apply safe
proof goal_cases
case (1 a b)
show ?case
using f_int[of a b] unfolding absolutely_integrable_on_def by auto
next
case prems: (2 e)
have "\<exists>y\<in>sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> ?S - e"
proof (rule ccontr)
assume "\<not> ?thesis"
then have "?S \<le> ?S - e"
by (intro cSUP_least[OF D(1)]) auto
then show False
using prems by auto
qed
then obtain K where *: "\<exists>x\<in>{d. d division_of \<Union>d}. K = (\<Sum>k\<in>x. norm (integral k f))"
"SUPREMUM {d. d division_of \<Union>d} (sum (\<lambda>k. norm (integral k f))) - e < K"
by (auto simp add: image_iff not_le)
from this(1) obtain d where "d division_of \<Union>d"
and "K = (\<Sum>k\<in>d. norm (integral k f))"
by auto
note d = this(1) *(2)[unfolded this(2)]
note d'=division_ofD[OF this(1)]
have "bounded (\<Union>d)"
by (rule elementary_bounded,fact)
from this[unfolded bounded_pos] obtain K where
K: "0 < K" "\<forall>x\<in>\<Union>d. norm x \<le> K" by auto
show ?case
apply (rule_tac x="K + 1" in exI)
apply safe
proof -
fix a b :: 'n
assume ab: "ball 0 (K + 1) \<subseteq> cbox a b"
have *: "\<forall>s s1. ?S - e < s1 \<and> s1 \<le> s \<and> s < ?S + e \<longrightarrow> \<bar>s - ?S\<bar> < e"
by arith
show "norm (integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) - ?S) < e"
unfolding real_norm_def
apply (rule *[rule_format])
apply safe
apply (rule d(2))
proof goal_cases
case 1
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> sum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
apply (intro sum_mono)
subgoal for k
using d'(4)[of k] f_int
by (auto simp: absolutely_integrable_on_def integral_norm_bound_integral)
done
also have "\<dots> = integral (\<Union>d) (\<lambda>x. norm (f x))"
apply (rule integral_combine_division_bottomup[symmetric])
apply (rule d)
unfolding forall_in_division[OF d(1)]
using f_int unfolding absolutely_integrable_on_def
apply auto
done
also have "\<dots> \<le> integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)"
proof -
have "\<Union>d \<subseteq> cbox a b"
apply rule
apply (drule K(2)[rule_format])
apply (rule ab[unfolded subset_eq,rule_format])
apply (auto simp add: dist_norm)
done
then show ?thesis
apply -
apply (subst if_P)
apply rule
apply (rule integral_subset_le)
defer
apply (rule integrable_on_subdivision[of _ _ _ "cbox a b"])
apply (rule d)
using f_int[of a b] unfolding absolutely_integrable_on_def
apply auto
done
qed
finally show ?case .
next
note f' = f_int[of a b, unfolded absolutely_integrable_on_def]
note f = f'[THEN conjunct1] f'[THEN conjunct2]
note * = this(2)[unfolded has_integral_integral has_integral[of "\<lambda>x. norm (f x)"],rule_format]
have "e/2>0"
using \<open>e > 0\<close> by auto
from * [OF this] obtain d1 where
d1: "gauge d1" "\<forall>p. p tagged_division_of (cbox a b) \<and> d1 fine p \<longrightarrow>
norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e / 2"
by auto
from henstock_lemma [OF f(1) \<open>e/2>0\<close>] obtain d2 where
d2: "gauge d2" "\<forall>p. p tagged_partial_division_of (cbox a b) \<and> d2 fine p \<longrightarrow>
(\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2" .
obtain p where
p: "p tagged_division_of (cbox a b)" "d1 fine p" "d2 fine p"
by (rule fine_division_exists [OF gauge_Int [OF d1(1) d2(1)], of a b])
(auto simp add: fine_Int)
have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e / 2 \<longrightarrow>
\<bar>sf' - di\<bar> < e / 2 \<longrightarrow> di < ?S + e"
by arith
show "integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < ?S + e"
apply (subst if_P)
apply rule
proof (rule *[rule_format])
show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e / 2"
unfolding split_def
apply (rule helplemma)
using d2(2)[rule_format,of p]
using p(1,3)
unfolding tagged_division_of_def split_def
apply auto
done
show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e / 2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]]
by (simp only: real_norm_def)
show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
apply (rule sum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
apply (drule tagged_division_ofD(4)[OF p(1)])
unfolding norm_scaleR
apply (subst abs_of_nonneg)
apply auto
done
show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
using partial_division_of_tagged_division[of p "cbox a b"] p(1)
apply (subst sum.over_tagged_division_lemma[OF p(1)])
apply (simp add: content_eq_0_interior)
apply (intro cSUP_upper2[OF D(2), of "snd ` p"])
apply (auto simp: tagged_partial_division_of_def)
done
qed
qed
qed (insert K, auto)
qed
then show "(\<lambda>x. norm (f x)) integrable_on UNIV"
by blast
qed
lemma absolutely_integrable_add[intro]:
fixes f g :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
shows "f absolutely_integrable_on s \<Longrightarrow> g absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) absolutely_integrable_on s"
by (rule set_integral_add)
lemma absolutely_integrable_diff[intro]:
fixes f g :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
shows "f absolutely_integrable_on s \<Longrightarrow> g absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) absolutely_integrable_on s"
by (rule set_integral_diff)
lemma absolutely_integrable_linear:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
and h :: "'n::euclidean_space \<Rightarrow> 'p::euclidean_space"
shows "f absolutely_integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) absolutely_integrable_on s"
using integrable_bounded_linear[of h lebesgue "\<lambda>x. indicator s x *\<^sub>R f x"]
by (simp add: linear_simps[of h])
lemma absolutely_integrable_sum:
fixes f :: "'a \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "finite t" and "\<And>a. a \<in> t \<Longrightarrow> (f a) absolutely_integrable_on s"
shows "(\<lambda>x. sum (\<lambda>a. f a x) t) absolutely_integrable_on s"
using assms(1,2) by induct auto
lemma absolutely_integrable_integrable_bound:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes le: "\<forall>x\<in>s. norm (f x) \<le> g x" and f: "f integrable_on s" and g: "g integrable_on s"
shows "f absolutely_integrable_on s"
proof (rule Bochner_Integration.integrable_bound)
show "g absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
proof
show "(\<lambda>x. norm (g x)) integrable_on s"
using le norm_ge_zero[of "f _"]
by (intro integrable_spike_finite[OF _ _ g, of "{}"])
(auto intro!: abs_of_nonneg intro: order_trans simp del: norm_ge_zero)
qed fact
show "set_borel_measurable lebesgue s f"
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
qed (use le in \<open>auto intro!: always_eventually split: split_indicator\<close>)
subsection \<open>Componentwise\<close>
proposition absolutely_integrable_componentwise_iff:
shows "f absolutely_integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A)"
proof -
have *: "(\<lambda>x. norm (f x)) integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. norm (f x \<bullet> b)) integrable_on A)"
if "f integrable_on A"
proof -
have 1: "\<And>i. \<lbrakk>(\<lambda>x. norm (f x)) integrable_on A; i \<in> Basis\<rbrakk>
\<Longrightarrow> (\<lambda>x. f x \<bullet> i) absolutely_integrable_on A"
apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. norm(f x)"])
using Basis_le_norm integrable_component that apply fastforce+
done
have 2: "\<forall>i\<in>Basis. (\<lambda>x. \<bar>f x \<bullet> i\<bar>) integrable_on A \<Longrightarrow> f absolutely_integrable_on A"
apply (rule absolutely_integrable_integrable_bound [where g = "\<lambda>x. \<Sum>i\<in>Basis. norm (f x \<bullet> i)"])
using norm_le_l1 that apply (force intro: integrable_sum)+
done
show ?thesis
apply auto
apply (metis (full_types) absolutely_integrable_on_def set_integrable_abs 1)
apply (metis (full_types) absolutely_integrable_on_def 2)
done
qed
show ?thesis
unfolding absolutely_integrable_on_def
by (simp add: integrable_componentwise_iff [symmetric] ball_conj_distrib * cong: conj_cong)
qed
lemma absolutely_integrable_componentwise:
shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A) \<Longrightarrow> f absolutely_integrable_on A"
by (simp add: absolutely_integrable_componentwise_iff)
lemma absolutely_integrable_component:
"f absolutely_integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (b :: 'b :: euclidean_space)) absolutely_integrable_on A"
by (drule absolutely_integrable_linear[OF _ bounded_linear_inner_left[of b]]) (simp add: o_def)
lemma absolutely_integrable_scaleR_left:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S"
proof -
have "(\<lambda>x. c *\<^sub>R x) o f absolutely_integrable_on S"
apply (rule absolutely_integrable_linear [OF assms])
by (simp add: bounded_linear_scaleR_right)
then show ?thesis by simp
qed
lemma absolutely_integrable_scaleR_right:
assumes "f absolutely_integrable_on S"
shows "(\<lambda>x. f x *\<^sub>R c) absolutely_integrable_on S"
using assms by blast
lemma absolutely_integrable_norm:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(norm o f) absolutely_integrable_on S"
using assms unfolding absolutely_integrable_on_def by auto
lemma absolutely_integrable_abs:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) absolutely_integrable_on S"
(is "?g absolutely_integrable_on S")
proof -
have eq: "?g =
(\<lambda>x. \<Sum>i\<in>Basis. ((\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0) \<circ>
(\<lambda>x. norm(\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f) x)"
by (simp add: sum.delta)
have *: "(\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0) \<circ>
(\<lambda>x. norm (\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f
absolutely_integrable_on S"
if "i \<in> Basis" for i
proof -
have "bounded_linear (\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else 0)"
by (simp add: linear_linear algebra_simps linearI)
moreover have "(\<lambda>x. norm (\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else 0)) \<circ> f
absolutely_integrable_on S"
unfolding o_def
apply (rule absolutely_integrable_norm [unfolded o_def])
using assms \<open>i \<in> Basis\<close>
apply (auto simp: algebra_simps dest: absolutely_integrable_component[where b=i])
done
ultimately show ?thesis
by (subst comp_assoc) (blast intro: absolutely_integrable_linear)
qed
show ?thesis
apply (rule ssubst [OF eq])
apply (rule absolutely_integrable_sum)
apply (force simp: intro!: *)+
done
qed
lemma abs_absolutely_integrableI_1:
fixes f :: "'a :: euclidean_space \<Rightarrow> real"
assumes f: "f integrable_on A" and "(\<lambda>x. \<bar>f x\<bar>) integrable_on A"
shows "f absolutely_integrable_on A"
by (rule absolutely_integrable_integrable_bound [OF _ assms]) auto
lemma abs_absolutely_integrableI:
assumes f: "f integrable_on S" and fcomp: "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S"
shows "f absolutely_integrable_on S"
proof -
have "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S" if "i \<in> Basis" for i
proof -
have "(\<lambda>x. \<bar>f x \<bullet> i\<bar>) integrable_on S"
using assms integrable_component [OF fcomp, where y=i] that by simp
then have "(\<lambda>x. f x \<bullet> i) absolutely_integrable_on S"
apply -
apply (rule abs_absolutely_integrableI_1, auto)
by (simp add: f integrable_component)
then show ?thesis
by (rule absolutely_integrable_scaleR_right)
qed
then have "(\<lambda>x. \<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S"
by (simp add: absolutely_integrable_sum)
then show ?thesis
by (simp add: euclidean_representation)
qed
lemma absolutely_integrable_abs_iff:
"f absolutely_integrable_on S \<longleftrightarrow>
f integrable_on S \<and> (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using absolutely_integrable_abs absolutely_integrable_on_def by blast
next
assume ?rhs
moreover
have "(\<lambda>x. if x \<in> S then \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i else 0) = (\<lambda>x. \<Sum>i\<in>Basis. \<bar>(if x \<in> S then f x else 0) \<bullet> i\<bar> *\<^sub>R i)"
by force
ultimately show ?lhs
by (simp only: absolutely_integrable_restrict_UNIV [of S, symmetric] integrable_restrict_UNIV [of S, symmetric] abs_absolutely_integrableI)
qed
lemma absolutely_integrable_max:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(\<lambda>x. \<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i)
absolutely_integrable_on S"
proof -
have "(\<lambda>x. \<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
(\<lambda>x. (1/2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))"
proof (rule ext)
fix x
have "(\<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. ((f x \<bullet> i + g x \<bullet> i + \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) / 2) *\<^sub>R i)"
by (force intro: sum.cong)
also have "... = (1 / 2) *\<^sub>R (\<Sum>i\<in>Basis. (f x \<bullet> i + g x \<bullet> i + \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) *\<^sub>R i)"
by (simp add: scaleR_right.sum)
also have "... = (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))"
by (simp add: sum.distrib algebra_simps euclidean_representation)
finally
show "(\<Sum>i\<in>Basis. max (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
(1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))" .
qed
moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
absolutely_integrable_on S"
apply (intro absolutely_integrable_add absolutely_integrable_scaleR_left assms)
using absolutely_integrable_abs [OF absolutely_integrable_diff [OF assms]]
apply (simp add: algebra_simps)
done
ultimately show ?thesis by metis
qed
corollary absolutely_integrable_max_1:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(\<lambda>x. max (f x) (g x)) absolutely_integrable_on S"
using absolutely_integrable_max [OF assms] by simp
lemma absolutely_integrable_min:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(\<lambda>x. \<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i)
absolutely_integrable_on S"
proof -
have "(\<lambda>x. \<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
(\<lambda>x. (1/2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))"
proof (rule ext)
fix x
have "(\<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. ((f x \<bullet> i + g x \<bullet> i - \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) / 2) *\<^sub>R i)"
by (force intro: sum.cong)
also have "... = (1 / 2) *\<^sub>R (\<Sum>i\<in>Basis. (f x \<bullet> i + g x \<bullet> i - \<bar>f x \<bullet> i - g x \<bullet> i\<bar>) *\<^sub>R i)"
by (simp add: scaleR_right.sum)
also have "... = (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))"
by (simp add: sum.distrib sum_subtractf algebra_simps euclidean_representation)
finally
show "(\<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) =
(1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i))" .
qed
moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
absolutely_integrable_on S"
apply (intro absolutely_integrable_add absolutely_integrable_diff absolutely_integrable_scaleR_left assms)
using absolutely_integrable_abs [OF absolutely_integrable_diff [OF assms]]
apply (simp add: algebra_simps)
done
ultimately show ?thesis by metis
qed
corollary absolutely_integrable_min_1:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(\<lambda>x. min (f x) (g x)) absolutely_integrable_on S"
using absolutely_integrable_min [OF assms] by simp
lemma nonnegative_absolutely_integrable:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "f integrable_on A" and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> 0 \<le> f x \<bullet> b"
shows "f absolutely_integrable_on A"
proof -
have "(\<lambda>x. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on A" if "i \<in> Basis" for i
proof -
have "(\<lambda>x. f x \<bullet> i) integrable_on A"
by (simp add: assms(1) integrable_component)
then have "(\<lambda>x. f x \<bullet> i) absolutely_integrable_on A"
by (metis that comp nonnegative_absolutely_integrable_1)
then show ?thesis
by (rule absolutely_integrable_scaleR_right)
qed
then have "(\<lambda>x. \<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) absolutely_integrable_on A"
by (simp add: absolutely_integrable_sum)
then show ?thesis
by (simp add: euclidean_representation)
qed
lemma absolutely_integrable_component_ubound:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes f: "f integrable_on A" and g: "g absolutely_integrable_on A"
and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b"
shows "f absolutely_integrable_on A"
proof -
have "(\<lambda>x. g x - (g x - f x)) absolutely_integrable_on A"
apply (rule absolutely_integrable_diff [OF g nonnegative_absolutely_integrable])
using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
by (simp add: comp inner_diff_left)
then show ?thesis
by simp
qed
lemma absolutely_integrable_component_lbound:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes f: "f absolutely_integrable_on A" and g: "g integrable_on A"
and comp: "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b"
shows "g absolutely_integrable_on A"
proof -
have "(\<lambda>x. f x + (g x - f x)) absolutely_integrable_on A"
apply (rule absolutely_integrable_add [OF f nonnegative_absolutely_integrable])
using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
by (simp add: comp inner_diff_left)
then show ?thesis
by simp
qed
subsection \<open>Dominated convergence\<close>
lemma dominated_convergence:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "\<And>k. (f k) integrable_on s" and h: "h integrable_on s"
and le: "\<And>k. \<forall>x \<in> s. norm (f k x) \<le> h x"
and conv: "\<forall>x \<in> s. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
shows "g integrable_on s" "(\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
proof -
have 3: "h absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
proof
show "(\<lambda>x. norm (h x)) integrable_on s"
proof (intro integrable_spike_finite[OF _ _ h, of "{}"] ballI)
fix x assume "x \<in> s - {}" then show "norm (h x) = h x"
by (metis Diff_empty abs_of_nonneg bot_set_def le norm_ge_zero order_trans real_norm_def)
qed auto
qed fact
have 2: "set_borel_measurable lebesgue s (f k)" for k
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
then have 1: "set_borel_measurable lebesgue s g"
by (rule borel_measurable_LIMSEQ_metric) (use conv in \<open>auto split: split_indicator\<close>)
have 4: "AE x in lebesgue. (\<lambda>i. indicator s x *\<^sub>R f i x) \<longlonglongrightarrow> indicator s x *\<^sub>R g x"
"AE x in lebesgue. norm (indicator s x *\<^sub>R f k x) \<le> indicator s x *\<^sub>R h x" for k
using conv le by (auto intro!: always_eventually split: split_indicator)
have g: "g absolutely_integrable_on s"
using 1 2 3 4 by (rule integrable_dominated_convergence)
then show "g integrable_on s"
by (auto simp: absolutely_integrable_on_def)
have "(\<lambda>k. (LINT x:s|lebesgue. f k x)) \<longlonglongrightarrow> (LINT x:s|lebesgue. g x)"
using 1 2 3 4 by (rule integral_dominated_convergence)
then show "(\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
using g absolutely_integrable_integrable_bound[OF le f h]
by (subst (asm) (1 2) set_lebesgue_integral_eq_integral) auto
qed
lemma has_integral_dominated_convergence:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "\<And>k. (f k has_integral y k) s" "h integrable_on s"
"\<And>k. \<forall>x\<in>s. norm (f k x) \<le> h x" "\<forall>x\<in>s. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
and x: "y \<longlonglongrightarrow> x"
shows "(g has_integral x) s"
proof -
have int_f: "\<And>k. (f k) integrable_on s"
using assms by (auto simp: integrable_on_def)
have "(g has_integral (integral s g)) s"
by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
moreover have "integral s g = x"
proof (rule LIMSEQ_unique)
show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> x"
using integral_unique[OF assms(1)] x by simp
show "(\<lambda>i. integral s (f i)) \<longlonglongrightarrow> integral s g"
by (intro dominated_convergence[OF int_f assms(2)]) fact+
qed
ultimately show ?thesis
by simp
qed
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_lowerbound[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
lemma has_integral_0_closure_imp_0:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes f: "continuous_on (closure S) f"
and nonneg_interior: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
and pos: "0 < emeasure lborel S"
and finite: "emeasure lborel S < \<infinity>"
and regular: "emeasure lborel (closure S) = emeasure lborel S"
and opn: "open S"
assumes int: "(f has_integral 0) (closure S)"
assumes x: "x \<in> closure S"
shows "f x = 0"
proof -
have zero: "emeasure lborel (frontier S) = 0"
using finite closure_subset regular
unfolding frontier_def
by (subst emeasure_Diff) (auto simp: frontier_def interior_open \<open>open S\<close> )
have nonneg: "0 \<le> f x" if "x \<in> closure S" for x
using continuous_ge_on_closure[OF f that nonneg_interior] by simp
have "0 = integral (closure S) f"
by (blast intro: int sym)
also
note intl = has_integral_integrable[OF int]
have af: "f absolutely_integrable_on (closure S)"
using nonneg
by (intro absolutely_integrable_onI intl integrable_eq[OF _ intl]) simp
then have "integral (closure S) f = set_lebesgue_integral lebesgue (closure S) f"
by (intro set_lebesgue_integral_eq_integral(2)[symmetric])
also have "\<dots> = 0 \<longleftrightarrow> (AE x in lebesgue. indicator (closure S) x *\<^sub>R f x = 0)"
by (rule integral_nonneg_eq_0_iff_AE[OF af]) (use nonneg in \<open>auto simp: indicator_def\<close>)
also have "\<dots> \<longleftrightarrow> (AE x in lebesgue. x \<in> {x. x \<in> closure S \<longrightarrow> f x = 0})"
by (auto simp: indicator_def)
finally have "(AE x in lebesgue. x \<in> {x. x \<in> closure S \<longrightarrow> f x = 0})" by simp
moreover have "(AE x in lebesgue. x \<in> - frontier S)"
using zero
by (auto simp: eventually_ae_filter null_sets_def intro!: exI[where x="frontier S"])
ultimately have ae: "AE x \<in> S in lebesgue. x \<in> {x \<in> closure S. f x = 0}" (is ?th)
by eventually_elim (use closure_subset in \<open>auto simp: \<close>)
have "closed {0::real}" by simp
with continuous_on_closed_vimage[OF closed_closure, of S f] f
have "closed (f -` {0} \<inter> closure S)" by blast
then have "closed {x \<in> closure S. f x = 0}" by (auto simp: vimage_def Int_def conj_commute)
with \<open>open S\<close> have "x \<in> {x \<in> closure S. f x = 0}" if "x \<in> S" for x using ae that
by (rule mem_closed_if_AE_lebesgue_open)
then have "f x = 0" if "x \<in> S" for x using that by auto
from continuous_constant_on_closure[OF f this \<open>x \<in> closure S\<close>]
show "f x = 0" .
qed
lemma has_integral_0_cbox_imp_0:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes f: "continuous_on (cbox a b) f"
and nonneg_interior: "\<And>x. x \<in> box a b \<Longrightarrow> 0 \<le> f x"
assumes int: "(f has_integral 0) (cbox a b)"
assumes ne: "box a b \<noteq> {}"
assumes x: "x \<in> cbox a b"
shows "f x = 0"
proof -
have "0 < emeasure lborel (box a b)"
using ne x unfolding emeasure_lborel_box_eq
by (force intro!: prod_pos simp: mem_box algebra_simps)
then show ?thesis using assms
by (intro has_integral_0_closure_imp_0[of "box a b" f x])
(auto simp: emeasure_lborel_box_eq emeasure_lborel_cbox_eq algebra_simps mem_box)
qed
end