Tagged some theories in HOL-Analysis: Cauchy_Integral_Theorem, Riemann_Mapping and Winding_Numbers.
(* Title: HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
Huge cleanup by LCP
*)
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: top_unique simp flip: 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>(snd ` (p - s))"
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>(B ` UNIV)"
unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
then show "countable (range B)" "space lebesgue \<subseteq> \<Union>(B ` UNIV)"
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) 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>(F ` UNIV) \<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>(F ` UNIV) \<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 simp flip: 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 "\<And>k. U k integrable_on UNIV" using U_int by auto
show "\<And>k x. x\<in>UNIV \<Longrightarrow> U k x \<le> U (Suc k) x" using \<open>incseq U\<close> by (auto simp: incseq_def le_fun_def)
then show "bounded (range (\<lambda>k. integral UNIV (U k)))"
using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
show "\<And>x. x\<in>UNIV \<Longrightarrow> (\<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]
obtain F where F: "\<And>i. simple_function lborel (F i)" "incseq F"
"\<And>i x. F i x < top" "\<And>x. (SUP i. F i x) = ennreal (f x)"
by blast
then have [measurable]: "\<And>i. F i \<in> borel_measurable lborel"
by (metis borel_measurable_simple_function)
let ?B = "\<lambda>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One) :: 'a set"
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))"
unfolding set_lebesgue_integral_def
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"
unfolding set_integrable_def
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]
unfolding set_integrable_def 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"
using assms has_integral_integral_lborel
unfolding set_integrable_def set_lebesgue_integral_def by blast
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 f
by (fastforce simp add: set_integrable_def set_lebesgue_integral_def indicator_def if_distrib[of "\<lambda>x. x *\<^sub>R f _"] 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_zero [simp]: "(\<lambda>x. 0) absolutely_integrable_on S"
by (simp add: set_integrable_def)
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))"
using integrable_norm set_integrable_def by blast
show "f integrable_on S"
by (rule set_lebesgue_integral_eq_integral[OF f])
have "(\<lambda>x. norm (indicator S x *\<^sub>R f x)) = (\<lambda>x. if x \<in> S then norm (f x) else 0)"
by auto
with integrable_on_lebesgue[OF nf] show "(\<lambda>x. norm (f x)) integrable_on S"
by (simp 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"
unfolding set_integrable_def
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 integrable_on_lebesgue_on:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "integrable (lebesgue_on S) f" and S: "S \<in> sets lebesgue"
shows "f integrable_on S"
proof -
have "integrable lebesgue (\<lambda>x. indicator S x *\<^sub>R f x)"
using S f inf_top.comm_neutral integrable_restrict_space by blast
then show ?thesis
using absolutely_integrable_on_def set_integrable_def by blast
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"
unfolding set_integrable_def
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"
by (rule absolutely_integrable_onI [OF f]) (use assms in \<open>simp add: integrable_eq\<close>)
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 absolutely_integrable_on_scaleR_iff:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows
"(\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S \<longleftrightarrow>
c = 0 \<or> f absolutely_integrable_on S"
proof (cases "c=0")
case False
then show ?thesis
unfolding absolutely_integrable_on_def
by (simp add: norm_mult)
qed auto
lemma absolutely_integrable_spike:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f absolutely_integrable_on T" and S: "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
shows "g absolutely_integrable_on T"
using assms integrable_spike
unfolding absolutely_integrable_on_def by metis
lemma absolutely_integrable_negligible:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "negligible S"
shows "f absolutely_integrable_on S"
using assms by (simp add: absolutely_integrable_on_def integrable_negligible)
lemma absolutely_integrable_spike_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
shows "(f absolutely_integrable_on T \<longleftrightarrow> g absolutely_integrable_on T)"
using assms by (blast intro: absolutely_integrable_spike sym)
lemma absolutely_integrable_spike_set_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
shows "(f absolutely_integrable_on S \<longleftrightarrow> f absolutely_integrable_on T)"
proof -
have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV \<longleftrightarrow>
(\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
proof (rule absolutely_integrable_spike_eq)
show "negligible ({x \<in> S - T. f x \<noteq> 0} \<union> {x \<in> T - S. f x \<noteq> 0})"
by (rule negligible_Un [OF assms])
qed auto
with absolutely_integrable_restrict_UNIV show ?thesis
by blast
qed
lemma absolutely_integrable_spike_set:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f absolutely_integrable_on S" and neg: "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
shows "f absolutely_integrable_on T"
using absolutely_integrable_spike_set_eq f neg by blast
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 set_integrable_def)
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 (fastforce simp add: lmeasure_integral_UNIV indicator_def[abs_def] lmeasurable_iff_integrable_on)
lemma integrable_on_const: "S \<in> lmeasurable \<Longrightarrow> (\<lambda>x. c) integrable_on S"
unfolding lmeasurable_iff_integrable
by (metis (mono_tags, lifting) integrable_eq integrable_on_scaleR_left lmeasurable_iff_integrable lmeasurable_iff_integrable_on scaleR_one)
lemma integral_indicator:
assumes "(S \<inter> T) \<in> lmeasurable"
shows "integral T (indicator S) = measure lebesgue (S \<inter> T)"
proof -
have "integral UNIV (indicator (S \<inter> T)) = integral UNIV (\<lambda>a. if a \<in> S \<inter> T then 1::real else 0)"
by (meson indicator_def)
moreover
have "(indicator (S \<inter> T) has_integral measure lebesgue (S \<inter> T)) UNIV"
using assms by (simp add: lmeasurable_iff_has_integral)
ultimately have "integral UNIV (\<lambda>x. if x \<in> S \<inter> T then 1 else 0) = measure lebesgue (S \<inter> T)"
by (metis (no_types) integral_unique)
then show ?thesis
using integral_restrict_Int [of UNIV "S \<inter> T" "\<lambda>x. 1::real"]
apply (simp add: integral_restrict_Int [symmetric])
by (meson indicator_def)
qed
lemma measurable_integrable:
fixes S :: "'a::euclidean_space set"
shows "S \<in> lmeasurable \<longleftrightarrow> (indicat_real S) integrable_on UNIV"
by (auto simp: lmeasurable_iff_integrable absolutely_integrable_on_iff_nonneg [symmetric] set_integrable_def)
lemma integrable_on_indicator:
fixes S :: "'a::euclidean_space set"
shows "indicat_real S integrable_on T \<longleftrightarrow> (S \<inter> T) \<in> lmeasurable"
unfolding measurable_integrable
unfolding integrable_restrict_UNIV [of T, symmetric]
by (fastforce simp add: indicator_def elim: integrable_eq)
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
lemma has_measure_limit:
assumes "S \<in> lmeasurable" "e > 0"
obtains B where "B > 0"
"\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> \<bar>measure lebesgue (S \<inter> cbox a b) - measure lebesgue S\<bar> < e"
using assms unfolding lmeasurable_iff_has_integral has_integral_alt'
by (force simp: integral_indicator integrable_on_indicator)
lemma lmeasurable_iff_indicator_has_integral:
fixes S :: "'a::euclidean_space set"
shows "S \<in> lmeasurable \<and> m = measure lebesgue S \<longleftrightarrow> (indicat_real S has_integral m) UNIV"
by (metis has_integral_iff lmeasurable_iff_has_integral measurable_integrable)
lemma has_measure_limit_iff:
fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
shows "S \<in> lmeasurable \<and> m = measure lebesgue S \<longleftrightarrow>
(\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
(S \<inter> cbox a b) \<in> lmeasurable \<and> \<bar>measure lebesgue (S \<inter> cbox a b) - m\<bar> < e)" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (meson has_measure_limit fmeasurable.Int lmeasurable_cbox)
next
assume RHS [rule_format]: ?rhs
show ?lhs
apply (simp add: lmeasurable_iff_indicator_has_integral has_integral' [where i=m])
using RHS
by (metis (full_types) integral_indicator integrable_integral integrable_on_indicator)
qed
subsection\<open>Applications to Negligibility\<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 ((+) (-a) ` S)"
proof (subst negligible_on_intervals, intro allI)
fix u v
show "negligible ((+) (- 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
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_base 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 auto
then show ?thesis
proof cases
case 1
show ?thesis
by (rule negligible_subset [of "closure S"])
(simp_all add: 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)
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)
proposition open_not_negligible:
assumes "open S" "S \<noteq> {}"
shows "\<not> negligible S"
proof
assume neg: "negligible S"
obtain a where "a \<in> S"
using \<open>S \<noteq> {}\<close> by blast
then obtain e where "e > 0" "cball a e \<subseteq> S"
using \<open>open S\<close> open_contains_cball_eq by blast
let ?p = "a - (e / DIM('a)) *\<^sub>R One" let ?q = "a + (e / DIM('a)) *\<^sub>R One"
have "cbox ?p ?q \<subseteq> cball a e"
proof (clarsimp simp: mem_box dist_norm)
fix x
assume "\<forall>i\<in>Basis. ?p \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> ?q \<bullet> i"
then have ax: "\<bar>(a - x) \<bullet> i\<bar> \<le> e / real DIM('a)" if "i \<in> Basis" for i
using that by (auto simp: algebra_simps)
have "norm (a - x) \<le> (\<Sum>i\<in>Basis. \<bar>(a - x) \<bullet> i\<bar>)"
by (rule norm_le_l1)
also have "\<dots> \<le> DIM('a) * (e / real DIM('a))"
by (intro sum_bounded_above ax)
also have "\<dots> = e"
by simp
finally show "norm (a - x) \<le> e" .
qed
then have "negligible (cbox ?p ?q)"
by (meson \<open>cball a e \<subseteq> S\<close> neg negligible_subset)
with \<open>e > 0\<close> show False
by (simp add: negligible_interval box_eq_empty algebra_simps divide_simps mult_le_0_iff)
qed
lemma negligible_convex_interior:
"convex S \<Longrightarrow> (negligible S \<longleftrightarrow> interior S = {})"
apply safe
apply (metis interior_subset negligible_subset open_interior open_not_negligible)
apply (metis Diff_empty closure_subset frontier_def negligible_convex_frontier negligible_subset)
done
lemma measure_eq_0_null_sets: "S \<in> null_sets M \<Longrightarrow> measure M S = 0"
by (auto simp: measure_def null_sets_def)
lemma negligible_imp_measure0: "negligible S \<Longrightarrow> measure lebesgue S = 0"
by (simp add: measure_eq_0_null_sets negligible_iff_null_sets)
lemma negligible_iff_emeasure0: "S \<in> sets lebesgue \<Longrightarrow> (negligible S \<longleftrightarrow> emeasure lebesgue S = 0)"
by (auto simp: measure_eq_0_null_sets negligible_iff_null_sets)
lemma negligible_iff_measure0: "S \<in> lmeasurable \<Longrightarrow> (negligible S \<longleftrightarrow> measure lebesgue S = 0)"
apply (auto simp: measure_eq_0_null_sets negligible_iff_null_sets)
by (metis completion.null_sets_outer subsetI)
lemma negligible_imp_sets: "negligible S \<Longrightarrow> S \<in> sets lebesgue"
by (simp add: negligible_iff_null_sets null_setsD2)
lemma negligible_imp_measurable: "negligible S \<Longrightarrow> S \<in> lmeasurable"
by (simp add: fmeasurableI_null_sets negligible_iff_null_sets)
lemma negligible_iff_measure: "negligible S \<longleftrightarrow> S \<in> lmeasurable \<and> measure lebesgue S = 0"
by (fastforce simp: negligible_iff_measure0 negligible_imp_measurable dest: negligible_imp_measure0)
lemma negligible_outer:
"negligible S \<longleftrightarrow> (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T < e)" (is "_ = ?rhs")
proof
assume "negligible S" then show ?rhs
by (metis negligible_iff_measure order_refl)
next
assume ?rhs then show "negligible S"
by (meson completion.null_sets_outer negligible_iff_null_sets)
qed
lemma negligible_outer_le:
"negligible S \<longleftrightarrow> (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> measure lebesgue T \<le> e)" (is "_ = ?rhs")
proof
assume "negligible S" then show ?rhs
by (metis dual_order.strict_implies_order negligible_imp_measurable negligible_imp_measure0 order_refl)
next
assume ?rhs then show "negligible S"
by (metis le_less_trans negligible_outer field_lbound_gt_zero)
qed
lemma negligible_UNIV: "negligible S \<longleftrightarrow> (indicat_real S has_integral 0) UNIV" (is "_=?rhs")
proof
assume ?rhs
then show "negligible S"
apply (auto simp: negligible_def has_integral_iff integrable_on_indicator)
by (metis negligible integral_unique lmeasure_integral_UNIV negligible_iff_measure0)
qed (simp add: negligible)
lemma sets_negligible_symdiff:
"\<lbrakk>S \<in> sets lebesgue; negligible((S - T) \<union> (T - S))\<rbrakk> \<Longrightarrow> T \<in> sets lebesgue"
by (metis Diff_Diff_Int Int_Diff_Un inf_commute negligible_Un_eq negligible_imp_sets sets.Diff sets.Un)
lemma lmeasurable_negligible_symdiff:
"\<lbrakk>S \<in> lmeasurable; negligible((S - T) \<union> (T - S))\<rbrakk> \<Longrightarrow> T \<in> lmeasurable"
using integrable_spike_set_eq lmeasurable_iff_integrable_on by blast
lemma measure_Un3_negligible:
assumes meas: "S \<in> lmeasurable" "T \<in> lmeasurable" "U \<in> lmeasurable"
and neg: "negligible(S \<inter> T)" "negligible(S \<inter> U)" "negligible(T \<inter> U)" and V: "S \<union> T \<union> U = V"
shows "measure lebesgue V = measure lebesgue S + measure lebesgue T + measure lebesgue U"
proof -
have [simp]: "measure lebesgue (S \<inter> T) = 0"
using neg(1) negligible_imp_measure0 by blast
have [simp]: "measure lebesgue (S \<inter> U \<union> T \<inter> U) = 0"
using neg(2) neg(3) negligible_Un negligible_imp_measure0 by blast
have "measure lebesgue V = measure lebesgue (S \<union> T \<union> U)"
using V by simp
also have "\<dots> = measure lebesgue S + measure lebesgue T + measure lebesgue U"
by (simp add: measure_Un3 meas fmeasurable.Un Int_Un_distrib2)
finally show ?thesis .
qed
lemma measure_translate_add:
assumes meas: "S \<in> lmeasurable" "T \<in> lmeasurable"
and U: "S \<union> ((+)a ` T) = U" and neg: "negligible(S \<inter> ((+)a ` T))"
shows "measure lebesgue S + measure lebesgue T = measure lebesgue U"
proof -
have [simp]: "measure lebesgue (S \<inter> (+) a ` T) = 0"
using neg negligible_imp_measure0 by blast
have "measure lebesgue (S \<union> ((+)a ` T)) = measure lebesgue S + measure lebesgue T"
by (simp add: measure_Un3 meas measurable_translation measure_translation fmeasurable.Un)
then show ?thesis
using U by auto
qed
lemma measure_negligible_symdiff:
assumes S: "S \<in> lmeasurable"
and neg: "negligible (S - T \<union> (T - S))"
shows "measure lebesgue T = measure lebesgue S"
proof -
have "measure lebesgue (S - T) = 0"
using neg negligible_Un_eq negligible_imp_measure0 by blast
then show ?thesis
by (metis S Un_commute add.right_neutral lmeasurable_negligible_symdiff measure_Un2 neg negligible_Un_eq negligible_imp_measure0)
qed
lemma measure_closure:
assumes "bounded S" and neg: "negligible (frontier S)"
shows "measure lebesgue (closure S) = measure lebesgue S"
proof -
have "measure lebesgue (frontier S) = 0"
by (metis neg negligible_imp_measure0)
then show ?thesis
by (metis assms lmeasurable_iff_integrable_on eq_iff_diff_eq_0 has_integral_interior integrable_on_def integral_unique lmeasurable_interior lmeasure_integral measure_frontier)
qed
lemma measure_interior:
"\<lbrakk>bounded S; negligible(frontier S)\<rbrakk> \<Longrightarrow> measure lebesgue (interior S) = measure lebesgue S"
using measure_closure measure_frontier negligible_imp_measure0 by fastforce
lemma measurable_Jordan:
assumes "bounded S" and neg: "negligible (frontier S)"
shows "S \<in> lmeasurable"
proof -
have "closure S \<in> lmeasurable"
by (metis lmeasurable_closure \<open>bounded S\<close>)
moreover have "interior S \<in> lmeasurable"
by (simp add: lmeasurable_interior \<open>bounded S\<close>)
moreover have "interior S \<subseteq> S"
by (simp add: interior_subset)
ultimately show ?thesis
using assms by (metis (full_types) closure_subset completion.complete_sets_sandwich_fmeasurable measure_closure measure_interior)
qed
lemma measurable_convex: "\<lbrakk>convex S; bounded S\<rbrakk> \<Longrightarrow> S \<in> lmeasurable"
by (simp add: measurable_Jordan negligible_convex_frontier)
subsection\<open>Negligibility of image under non-injective linear map\<close>
lemma negligible_Union_nat:
assumes "\<And>n::nat. negligible(S n)"
shows "negligible(\<Union>n. S n)"
proof -
have "negligible (\<Union>m\<le>k. S m)" for k
using assms by blast
then have 0: "integral UNIV (indicat_real (\<Union>m\<le>k. S m)) = 0"
and 1: "(indicat_real (\<Union>m\<le>k. S m)) integrable_on UNIV" for k
by (auto simp: negligible has_integral_iff)
have 2: "\<And>k x. indicat_real (\<Union>m\<le>k. S m) x \<le> (indicat_real (\<Union>m\<le>Suc k. S m) x)"
by (simp add: indicator_def)
have 3: "\<And>x. (\<lambda>k. indicat_real (\<Union>m\<le>k. S m) x) \<longlonglongrightarrow> (indicat_real (\<Union>n. S n) x)"
by (force simp: indicator_def eventually_sequentially intro: Lim_eventually)
have 4: "bounded (range (\<lambda>k. integral UNIV (indicat_real (\<Union>m\<le>k. S m))))"
by (simp add: 0 image_def)
have *: "indicat_real (\<Union>n. S n) integrable_on UNIV \<and>
(\<lambda>k. integral UNIV (indicat_real (\<Union>m\<le>k. S m))) \<longlonglongrightarrow> (integral UNIV (indicat_real (\<Union>n. S n)))"
by (intro monotone_convergence_increasing 1 2 3 4)
then have "integral UNIV (indicat_real (\<Union>n. S n)) = (0::real)"
using LIMSEQ_unique by (auto simp: 0)
then show ?thesis
using * by (simp add: negligible_UNIV has_integral_iff)
qed
lemma negligible_linear_singular_image:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "linear f" "\<not> inj f"
shows "negligible (f ` S)"
proof -
obtain a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
using assms linear_singular_image_hyperplane by blast
then show "negligible (f ` S)"
by (metis negligible_hyperplane negligible_subset)
qed
lemma measure_negligible_finite_Union:
assumes "finite \<F>"
and meas: "\<And>S. S \<in> \<F> \<Longrightarrow> S \<in> lmeasurable"
and djointish: "pairwise (\<lambda>S T. negligible (S \<inter> T)) \<F>"
shows "measure lebesgue (\<Union>\<F>) = (\<Sum>S\<in>\<F>. measure lebesgue S)"
using assms
proof (induction)
case empty
then show ?case
by auto
next
case (insert S \<F>)
then have "S \<in> lmeasurable" "\<Union>\<F> \<in> lmeasurable" "pairwise (\<lambda>S T. negligible (S \<inter> T)) \<F>"
by (simp_all add: fmeasurable.finite_Union insert.hyps(1) insert.prems(1) pairwise_insert subsetI)
then show ?case
proof (simp add: measure_Un3 insert)
have *: "\<And>T. T \<in> (\<inter>) S ` \<F> \<Longrightarrow> negligible T"
using insert by (force simp: pairwise_def)
have "negligible(S \<inter> \<Union>\<F>)"
unfolding Int_Union
by (rule negligible_Union) (simp_all add: * insert.hyps(1))
then show "measure lebesgue (S \<inter> \<Union>\<F>) = 0"
using negligible_imp_measure0 by blast
qed
qed
lemma measure_negligible_finite_Union_image:
assumes "finite S"
and meas: "\<And>x. x \<in> S \<Longrightarrow> f x \<in> lmeasurable"
and djointish: "pairwise (\<lambda>x y. negligible (f x \<inter> f y)) S"
shows "measure lebesgue (\<Union>(f ` S)) = (\<Sum>x\<in>S. measure lebesgue (f x))"
proof -
have "measure lebesgue (\<Union>(f ` S)) = sum (measure lebesgue) (f ` S)"
using assms by (auto simp: pairwise_mono pairwise_image intro: measure_negligible_finite_Union)
also have "\<dots> = sum (measure lebesgue \<circ> f) S"
using djointish [unfolded pairwise_def] by (metis inf.idem negligible_imp_measure0 sum.reindex_nontrivial [OF \<open>finite S\<close>])
also have "\<dots> = (\<Sum>x\<in>S. measure lebesgue (f x))"
by simp
finally show ?thesis .
qed
subsection \<open>Negligibility of a Lipschitz image of a negligible set\<close>
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)
then have "S \<in> sets lebesgue"
by blast
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 - S) \<in> lmeasurable"
"measure lebesgue (T - S) < e/2 / (2 * B * DIM('M)) ^ DIM('N)"
by (rule lmeasurable_outer_open [OF \<open>S \<in> sets lebesgue\<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: measure_Diff_null_set negligible_iff_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)"
apply (rule prod.cong [OF refl])
by (simp add: \<open>X \<in> \<D>\<close> inner_diff_left prj1_idem)
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)
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" .
show "T \<in> lmeasurable"
using \<open>S \<in> lmeasurable\<close> \<open>S \<subseteq> T\<close> \<open>T - S \<in> lmeasurable\<close> fmeasurable_Diff_D by blast
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)
show "(\<Union>D\<in>\<D>. fbx D) \<in> lmeasurable"
by (intro fmeasurable_UN_bound[OF \<open>countable \<D>\<close> 1 2])
have "measure lebesgue (\<Union>D\<in>\<D>. fbx D) \<le> e/2"
by (intro measure_UN_bound[OF \<open>countable \<D>\<close> 1 2])
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
subsection\<open>Measurability of countable unions and intersections of various kinds.\<close>
lemma
assumes S: "\<And>n. S n \<in> lmeasurable"
and leB: "\<And>n. measure lebesgue (S n) \<le> B"
and nest: "\<And>n. S n \<subseteq> S(Suc n)"
shows measurable_nested_Union: "(\<Union>n. S n) \<in> lmeasurable"
and measure_nested_Union: "(\<lambda>n. measure lebesgue (S n)) \<longlonglongrightarrow> measure lebesgue (\<Union>n. S n)" (is ?Lim)
proof -
have 1: "\<And>n. (indicat_real (S n)) integrable_on UNIV"
using S measurable_integrable by blast
have 2: "\<And>n x::'a. indicat_real (S n) x \<le> (indicat_real (S (Suc n)) x)"
by (simp add: indicator_leI nest rev_subsetD)
have 3: "\<And>x. (\<lambda>n. indicat_real (S n) x) \<longlonglongrightarrow> (indicat_real (\<Union>(S ` UNIV)) x)"
apply (rule Lim_eventually)
apply (simp add: indicator_def)
by (metis eventually_sequentiallyI lift_Suc_mono_le nest subsetCE)
have 4: "bounded (range (\<lambda>n. integral UNIV (indicat_real (S n))))"
using leB by (auto simp: lmeasure_integral_UNIV [symmetric] S bounded_iff)
have "(\<Union>n. S n) \<in> lmeasurable \<and> ?Lim"
apply (simp add: lmeasure_integral_UNIV S cong: conj_cong)
apply (simp add: measurable_integrable)
apply (rule monotone_convergence_increasing [OF 1 2 3 4])
done
then show "(\<Union>n. S n) \<in> lmeasurable" "?Lim"
by auto
qed
lemma
assumes S: "\<And>n. S n \<in> lmeasurable"
and djointish: "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
and leB: "\<And>n. (\<Sum>k\<le>n. measure lebesgue (S k)) \<le> B"
shows measurable_countable_negligible_Union: "(\<Union>n. S n) \<in> lmeasurable"
and measure_countable_negligible_Union: "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. S n)" (is ?Sums)
proof -
have 1: "\<Union> (S ` {..n}) \<in> lmeasurable" for n
using S by blast
have 2: "measure lebesgue (\<Union> (S ` {..n})) \<le> B" for n
proof -
have "measure lebesgue (\<Union> (S ` {..n})) \<le> (\<Sum>k\<le>n. measure lebesgue (S k))"
by (simp add: S fmeasurableD measure_UNION_le)
with leB show ?thesis
using order_trans by blast
qed
have 3: "\<And>n. \<Union> (S ` {..n}) \<subseteq> \<Union> (S ` {..Suc n})"
by (simp add: SUP_subset_mono)
have eqS: "(\<Union>n. S n) = (\<Union>n. \<Union> (S ` {..n}))"
using atLeastAtMost_iff by blast
also have "(\<Union>n. \<Union> (S ` {..n})) \<in> lmeasurable"
by (intro measurable_nested_Union [OF 1 2] 3)
finally show "(\<Union>n. S n) \<in> lmeasurable" .
have eqm: "(\<Sum>i\<le>n. measure lebesgue (S i)) = measure lebesgue (\<Union> (S ` {..n}))" for n
using assms by (simp add: measure_negligible_finite_Union_image pairwise_mono)
have "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. \<Union> (S ` {..n}))"
by (simp add: sums_def' eqm atLeast0AtMost) (intro measure_nested_Union [OF 1 2] 3)
then show ?Sums
by (simp add: eqS)
qed
lemma negligible_countable_Union [intro]:
assumes "countable \<F>" and meas: "\<And>S. S \<in> \<F> \<Longrightarrow> negligible S"
shows "negligible (\<Union>\<F>)"
proof (cases "\<F> = {}")
case False
then show ?thesis
by (metis from_nat_into range_from_nat_into assms negligible_Union_nat)
qed simp
lemma
assumes S: "\<And>n. (S n) \<in> lmeasurable"
and djointish: "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
and bo: "bounded (\<Union>n. S n)"
shows measurable_countable_negligible_Union_bounded: "(\<Union>n. S n) \<in> lmeasurable"
and measure_countable_negligible_Union_bounded: "(\<lambda>n. (measure lebesgue (S n))) sums measure lebesgue (\<Union>n. S n)" (is ?Sums)
proof -
obtain a b where ab: "(\<Union>n. S n) \<subseteq> cbox a b"
using bo bounded_subset_cbox_symmetric by metis
then have B: "(\<Sum>k\<le>n. measure lebesgue (S k)) \<le> measure lebesgue (cbox a b)" for n
proof -
have "(\<Sum>k\<le>n. measure lebesgue (S k)) = measure lebesgue (\<Union> (S ` {..n}))"
using measure_negligible_finite_Union_image [OF _ _ pairwise_subset] djointish
by (metis S finite_atMost subset_UNIV)
also have "\<dots> \<le> measure lebesgue (cbox a b)"
apply (rule measure_mono_fmeasurable)
using ab S by force+
finally show ?thesis .
qed
show "(\<Union>n. S n) \<in> lmeasurable"
by (rule measurable_countable_negligible_Union [OF S djointish B])
show ?Sums
by (rule measure_countable_negligible_Union [OF S djointish B])
qed
lemma measure_countable_Union_approachable:
assumes "countable \<D>" "e > 0" and measD: "\<And>d. d \<in> \<D> \<Longrightarrow> d \<in> lmeasurable"
and B: "\<And>D'. \<lbrakk>D' \<subseteq> \<D>; finite D'\<rbrakk> \<Longrightarrow> measure lebesgue (\<Union>D') \<le> B"
obtains D' where "D' \<subseteq> \<D>" "finite D'" "measure lebesgue (\<Union>\<D>) - e < measure lebesgue (\<Union>D')"
proof (cases "\<D> = {}")
case True
then show ?thesis
by (simp add: \<open>e > 0\<close> that)
next
case False
let ?S = "\<lambda>n. \<Union>k \<le> n. from_nat_into \<D> k"
have "(\<lambda>n. measure lebesgue (?S n)) \<longlonglongrightarrow> measure lebesgue (\<Union>n. ?S n)"
proof (rule measure_nested_Union)
show "?S n \<in> lmeasurable" for n
by (simp add: False fmeasurable.finite_UN from_nat_into measD)
show "measure lebesgue (?S n) \<le> B" for n
by (metis (mono_tags, lifting) B False finite_atMost finite_imageI from_nat_into image_iff subsetI)
show "?S n \<subseteq> ?S (Suc n)" for n
by force
qed
then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> dist (measure lebesgue (?S n)) (measure lebesgue (\<Union>n. ?S n)) < e"
using metric_LIMSEQ_D \<open>e > 0\<close> by blast
show ?thesis
proof
show "from_nat_into \<D> ` {..N} \<subseteq> \<D>"
by (auto simp: False from_nat_into)
have eq: "(\<Union>n. \<Union>k\<le>n. from_nat_into \<D> k) = (\<Union>\<D>)"
using \<open>countable \<D>\<close> False
by (auto intro: from_nat_into dest: from_nat_into_surj [OF \<open>countable \<D>\<close>])
show "measure lebesgue (\<Union>\<D>) - e < measure lebesgue (\<Union> (from_nat_into \<D> ` {..N}))"
using N [OF order_refl]
by (auto simp: eq algebra_simps dist_norm)
qed auto
qed
subsection\<open>Negligibility is a local property\<close>
lemma locally_negligible_alt:
"negligible S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>U. openin (subtopology euclidean S) U \<and> x \<in> U \<and> negligible U)"
(is "_ = ?rhs")
proof
assume "negligible S"
then show ?rhs
using openin_subtopology_self by blast
next
assume ?rhs
then obtain U where ope: "\<And>x. x \<in> S \<Longrightarrow> openin (subtopology euclidean S) (U x)"
and cov: "\<And>x. x \<in> S \<Longrightarrow> x \<in> U x"
and neg: "\<And>x. x \<in> S \<Longrightarrow> negligible (U x)"
by metis
obtain \<F> where "\<F> \<subseteq> U ` S" "countable \<F>" and eq: "\<Union>\<F> = \<Union>(U ` S)"
using ope by (force intro: Lindelof_openin [of "U ` S" S])
then have "negligible (\<Union>\<F>)"
by (metis imageE neg negligible_countable_Union subset_eq)
with eq have "negligible (\<Union>(U ` S))"
by metis
moreover have "S \<subseteq> \<Union>(U ` S)"
using cov by blast
ultimately show "negligible S"
using negligible_subset by blast
qed
lemma locally_negligible:
"locally negligible S \<longleftrightarrow> negligible S"
unfolding locally_def
apply safe
apply (meson negligible_subset openin_subtopology_self locally_negligible_alt)
by (meson negligible_subset openin_imp_subset order_refl)
subsection\<open>Integral bounds\<close>
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 add: set_lebesgue_integral_def)
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 add: set_lebesgue_integral_def)
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 f by (force simp add: set_integrable_def)
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 absdiff_norm_less:
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
proposition bounded_variation_absolutely_integrable_interval:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
and *: "\<And>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\<in>?D. ?f x"
have *: "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>p. p tagged_division_of cbox a b \<and>
\<gamma> fine p \<longrightarrow>
norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e)"
if e: "e > 0" for e
proof -
have "?S - e/2 < ?S" using \<open>e > 0\<close> 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 "\<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}" for x
proof -
have "\<exists>d'>0. \<forall>x'\<in>\<Union>{i \<in> d. x \<notin> i}. d' \<le> dist x x'"
proof (rule separate_point_closed)
show "closed (\<Union>{i \<in> d. x \<notin> i})"
using d' by force
show "x \<notin> \<Union>{i \<in> d. x \<notin> i}"
by auto
qed
then show ?thesis
by force
qed
then obtain k where k: "\<And>x. 0 < k x" "\<And>i x. \<lbrakk>i \<in> d; x \<notin> i\<rbrakk> \<Longrightarrow> ball x (k x) \<inter> i = {}"
by metis
have "e/2 > 0"
using e by auto
with Henstock_lemma[OF f]
obtain \<gamma> where g: "gauge \<gamma>"
"\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk>
\<Longrightarrow> (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
by (metis (no_types, lifting))
let ?g = "\<lambda>x. \<gamma> x \<inter> ball x (k x)"
show ?thesis
proof (intro exI conjI allI impI)
show "gauge ?g"
using g(1) k(1) by (auto simp: gauge_def)
next
fix p
assume "p tagged_division_of (cbox a b) \<and> ?g fine p"
then have p: "p tagged_division_of cbox a b" "\<gamma> fine p" "(\<lambda>x. ball x (k x)) fine p"
by (auto simp: 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': "\<gamma> fine p'"
using p(2) by (auto simp: p'_def fine_def)
have p'': "p' tagged_division_of (cbox a b)"
proof (rule tagged_division_ofI)
show "finite p'"
proof (rule finite_subset)
show "p' \<subseteq> (\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p)"
by (force simp: p'_def image_iff)
show "finite ((\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p))"
by (simp add: d'(1) p'(1))
qed
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
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)] by (meson Int_interval)
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)] by (auto simp: il1 il2)
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 blast
have "y \<in> \<Union>{K. \<exists>x. (x, K) \<in> p'}" if y: "y \<in> cbox a b" for y
proof -
obtain x l where xl: "(x, l) \<in> p" "y \<in> l"
using y unfolding p'(6)[symmetric] by auto
obtain i where i: "i \<in> d" "y \<in> i"
using y unfolding d'(6)[symmetric] by auto
have "x \<in> i"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show ?thesis
unfolding p'_def by (rule_tac X="i \<inter> l" in UnionI) (use i xl in auto)
qed
show "\<Union>{K. \<exists>x. (x, K) \<in> p'} = cbox a b"
proof
show "\<Union>{k. \<exists>x. (x, k) \<in> p'} \<subseteq> cbox a b"
using * by auto
next
show "cbox a b \<subseteq> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
proof
fix y
assume y: "y \<in> cbox a b"
obtain x L where xl: "(x, L) \<in> p" "y \<in> L"
using y unfolding p'(6)[symmetric] by auto
obtain I where i: "I \<in> d" "y \<in> I"
using y unfolding d'(6)[symmetric] by auto
have "x \<in> I"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
apply (rule_tac X="I \<inter> L" in UnionI)
using i xl by (auto simp: p'_def)
qed
qed
qed
then have sum_less_e2: "(\<Sum>(x,K) \<in> p'. norm (content K *\<^sub>R f x - integral K f)) < e/2"
using g(2) gp' tagged_division_of_def by blast
have "(x, I \<inter> L) \<in> p'" if x: "(x, L) \<in> p" "I \<in> d" and y: "y \<in> I" "y \<in> L"
for x I L y
proof -
have "x \<in> I"
using fineD[OF p(3) that(1)] k(2)[OF \<open>I \<in> d\<close>] y by auto
with x have "(\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> I \<inter> L = i \<inter> l)"
by blast
then have "(x, I \<inter> L) \<in> p'"
by (simp add: p'_def)
with y show ?thesis by auto
qed
moreover have "\<exists>y i l. (x, K) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
if xK: "(x,K) \<in> p'" for x K
proof -
obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l"
using xK unfolding p'_def by auto
then show ?thesis
using p'(2) by fastforce
qed
ultimately have p'alt: "p' = {(x, I \<inter> L) | x I L. (x,L) \<in> p \<and> I \<in> d \<and> I \<inter> L \<noteq> {}}"
by auto
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
have snd_p_div: "snd ` p division_of cbox a b"
by (rule division_of_tagged_division[OF p(1)])
note snd_p = division_ofD[OF snd_p_div]
have fin_d_sndp: "finite (d \<times> snd ` p)"
by (simp add: d'(1) snd_p(1))
have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e/2; ?S - e/2 < sni; sni' \<le> ?S;
sni \<le> sni'; sf' = sf\<rbrakk> \<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
proof (rule *)
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"
using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
show "(\<Sum>(x,k) \<in> p'. norm (integral k f)) \<le>?S"
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = (\<lambda>(k,l). k \<inter> l) ` (d \<times> 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)
fix K assume k: "K \<in> d"
from d'(4)[OF this] obtain u v where uv: "K = cbox u v" by metis
define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}"
have uvab: "cbox u v \<subseteq> cbox a b"
using d(1) k uv by blast
have "d' division_of cbox u v"
unfolding d'_def by (rule division_inter_1 [OF snd_p_div uvab])
moreover then have "norm (\<Sum>i\<in>d'. integral i f) \<le> (\<Sum>k\<in>d'. norm (integral k f))"
by (simp add: sum_norm_le)
ultimately have "norm (integral K f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
apply (subst integral_combine_division_topdown[of _ _ d'])
apply (auto simp: uv intro: integrable_on_subcbox[OF assms(1) uvab])
done
also have "\<dots> = (\<Sum>I\<in>{K \<inter> L |L. L \<in> snd ` p}. norm (integral I f))"
proof -
have *: "norm (integral I f) = 0"
if "I \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
"I \<notin> {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}" for I
using that by auto
show ?thesis
apply (rule sum.mono_neutral_left)
apply (simp add: snd_p(1))
unfolding d'_def uv using * by auto
qed
also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))"
proof -
have *: "norm (integral (K \<inter> l) f) = 0"
if "l \<in> snd ` p" "y \<in> snd ` p" "l \<noteq> y" "K \<inter> l = K \<inter> y" for l y
proof -
have "interior (K \<inter> l) \<subseteq> interior (l \<inter> y)"
by (metis Int_lower2 interior_mono le_inf_iff that(4))
then have "interior (K \<inter> l) = {}"
by (simp add: snd_p(5) that)
moreover from d'(4)[OF k] snd_p(4)[OF that(1)]
obtain u1 v1 u2 v2
where uv: "K = cbox u1 u2" "l = cbox v1 v2" by metis
ultimately show ?thesis
using that integral_null
unfolding uv Int_interval content_eq_0_interior
by (metis (mono_tags, lifting) norm_eq_zero)
qed
show ?thesis
unfolding Setcompr_eq_image
apply (rule sum.reindex_nontrivial [unfolded o_def])
apply (rule finite_imageI)
apply (rule p')
using * by auto
qed
finally show "norm (integral K f) \<le> (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))" .
qed
also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
by (simp add: sum.cartesian_product)
also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod (\<inter>) x) f))"
by (force simp: split_def intro!: sum.cong)
also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
proof -
have eq0: " (integral (l1 \<inter> k1) f) = 0"
if "l1 \<inter> k1 = l2 \<inter> k2" "(l1, k1) \<noteq> (l2, k2)"
"l1 \<in> d" "(j1,k1) \<in> p" "l2 \<in> d" "(j2,k2) \<in> p"
for l1 l2 k1 k2 j1 j2
proof -
obtain u1 v1 u2 v2 where uv: "l1 = cbox u1 u2" "k1 = cbox v1 v2"
using \<open>(j1, k1) \<in> p\<close> \<open>l1 \<in> d\<close> d'(4) p'(4) by blast
have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
using that by auto
then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
by (meson d'(5) old.prod.inject p'(5) that(3) that(4) that(5) that(6))
moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
by (simp add: that(1))
ultimately have "interior(l1 \<inter> k1) = {}"
by auto
then show ?thesis
unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
qed
show ?thesis
unfolding *
apply (rule sum.reindex_nontrivial [OF fin_d_sndp, symmetric, unfolded o_def])
apply clarsimp
by (metis eq0 fst_conv snd_conv)
qed
also have "\<dots> = (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
have 0: "integral (ia \<inter> snd (a, b)) f = 0"
if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
proof -
have "ia \<inter> b = {}"
using that unfolding p'alt image_iff Bex_def not_ex
apply (erule_tac x="(a, ia \<inter> b)" in allE)
apply auto
done
then show ?thesis by auto
qed
have 1: "\<exists>i l. snd (a, b) = i \<inter> l \<and> i \<in> d \<and> l \<in> snd ` p" if "(a, b) \<in> p'" for a b
using that
apply (clarsimp simp: p'_def image_iff)
by (metis (no_types, hide_lams) snd_conv)
show ?thesis
unfolding sum_p'
apply (rule sum.mono_neutral_right)
apply (metis * finite_imageI[OF fin_d_sndp])
using 0 1 by auto
qed
finally show ?thesis .
qed
show "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
by force
have fin_pd: "finite (p \<times> d)"
using finite_cartesian_product[OF p'(1) d'(1)] by metis
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 [OF fin_pd])
unfolding p'alt apply auto
by fastforce
also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
proof -
have "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
if "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
"x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "x1 \<noteq> x2 \<or> l1 \<noteq> l2 \<or> k1 \<noteq> k2"
for x1 l1 k1 x2 l2 k2
proof -
obtain u1 v1 u2 v2 where uv: "k1 = cbox u1 u2" "l1 = cbox v1 v2"
by (meson \<open>(x1, l1) \<in> p\<close> \<open>k1 \<in> d\<close> d(1) division_ofD(4) p'(4))
have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
using that by auto
then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
apply (rule disjE)
using that p'(5) d'(5) by auto
moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
unfolding that ..
ultimately have "interior (l1 \<inter> k1) = {}"
by auto
then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
qed
then show ?thesis
unfolding *
apply (subst sum.reindex_nontrivial [OF fin_pd])
unfolding split_paired_all o_def split_def prod.inject
apply force+
done
qed
also have "\<dots> = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
have sumeq: "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
if "(x, l) \<in> p" for x l
proof -
note xl = p'(2-4)[OF that]
then obtain u v where uv: "l = cbox u v" by blast
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}"
proof -
have eq0: "content (k \<inter> cbox u v) = 0"
if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
proof -
from d'(4)[OF that(1)] d'(4)[OF that(2)]
obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
by (meson Int_interval)
have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
by (simp add: d'(5) that)
also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
by auto
also have "\<dots> = interior (k \<inter> cbox u v)"
unfolding eq by auto
finally show ?thesis
unfolding \<alpha> content_eq_0_interior ..
qed
then show ?thesis
unfolding Setcompr_eq_image
apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
by auto
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 [OF \<open>finite d\<close>])
apply (fastforce simp: inf.commute)+
done
finally show "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * 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
show ?thesis
by (subst sum_Sigma_product[symmetric]) (auto intro!: sumeq sum.cong p' d')
qed
finally show ?thesis .
qed
qed (rule d)
qed
qed
then show ?thesis
using absolutely_integrable_onI [OF f has_integral_integrable] has_integral[of _ ?S]
by blast
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\<in>?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 d K where ddiv: "d division_of \<Union>d" and "K = (\<Sum>k\<in>d. norm (integral k f))"
"Sup (sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union> d}) - e < K"
by (auto simp add: image_iff not_le)
then have d: "Sup (sum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union> d}) - e
< (\<Sum>k\<in>d. norm (integral k f))"
by auto
note d'=division_ofD[OF ddiv]
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
proof (intro conjI impI allI exI)
fix a b :: 'n
assume ab: "ball 0 (K + 1) \<subseteq> cbox a b"
have *: "\<And>s s1. \<lbrakk>?S - e < s1; s1 \<le> s; s < ?S + e\<rbrakk> \<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
proof (rule * [OF d])
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> sum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
proof (intro sum_mono)
fix k assume "k \<in> d"
with d'(4) f_int show "norm (integral k f) \<le> integral k (\<lambda>x. norm (f x))"
by (force simp: absolutely_integrable_on_def integral_norm_bound_integral)
qed
also have "\<dots> = integral (\<Union>d) (\<lambda>x. norm (f x))"
apply (rule integral_combine_division_bottomup[OF ddiv, symmetric])
using absolutely_integrable_on_def d'(4) f_int by blast
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"
using K(2) ab by fastforce
then show ?thesis
using integrable_on_subdivision[OF ddiv] f_int[of a b] unfolding absolutely_integrable_on_def
by (auto intro!: integral_subset_le)
qed
finally show "(\<Sum>k\<in>d. norm (integral k f))
\<le> integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)" .
next
have "e/2>0"
using \<open>e > 0\<close> by auto
moreover
have f: "f integrable_on cbox a b" "(\<lambda>x. norm (f x)) integrable_on cbox a b"
using f_int by (auto simp: absolutely_integrable_on_def)
ultimately obtain d1 where "gauge d1"
and d1: "\<And>p. \<lbrakk>p tagged_division_of (cbox a b); d1 fine p\<rbrakk> \<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"
unfolding has_integral_integral has_integral by meson
obtain d2 where "gauge d2"
and d2: "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox a b); d2 fine p\<rbrakk> \<Longrightarrow>
(\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
by (blast intro: Henstock_lemma [OF f(1) \<open>e/2>0\<close>])
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 \<open>gauge d1\<close> \<open>gauge d2\<close>], of a b])
(auto simp add: fine_Int)
have *: "\<And>sf sf' si di. \<lbrakk>sf' = sf; si \<le> ?S; \<bar>sf - si\<bar> < e/2;
\<bar>sf' - di\<bar> < e/2\<rbrakk> \<Longrightarrow> di < ?S + e"
by arith
have "integral (cbox a b) (\<lambda>x. norm (f x)) < ?S + e"
proof (rule *)
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 absdiff_norm_less)
using d2[of p] p(1,3) apply (auto simp: tagged_division_of_def split_def)
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[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))"
by (auto simp: split_paired_all sum.cong [OF refl])
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 (auto simp: content_eq_0_interior tagged_partial_division_of_def intro!: cSUP_upper2 D)
done
qed
then show "integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < ?S + e"
by simp
qed
qed (insert K, auto)
qed
then show "(\<lambda>x. norm (f x)) integrable_on UNIV"
by blast
qed
subsection\<open>Outer and inner approximation of measurable sets by well-behaved sets.\<close>
proposition measurable_outer_intervals_bounded:
assumes "S \<in> lmeasurable" "S \<subseteq> cbox a b" "e > 0"
obtains \<D>
where "countable \<D>"
"\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
"pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
"\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i)/2^n"
"\<And>K. \<lbrakk>K \<in> \<D>; box a b \<noteq> {}\<rbrakk> \<Longrightarrow> interior K \<noteq> {}"
"S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable" "measure lebesgue (\<Union>\<D>) \<le> measure lebesgue S + e"
proof (cases "box a b = {}")
case True
show ?thesis
proof (cases "cbox a b = {}")
case True
with assms have [simp]: "S = {}"
by auto
show ?thesis
proof
show "countable {}"
by simp
qed (use \<open>e > 0\<close> in auto)
next
case False
show ?thesis
proof
show "countable {cbox a b}"
by simp
show "\<And>u v. cbox u v \<in> {cbox a b} \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i)/2 ^ n"
using False by (force simp: eq_cbox intro: exI [where x=0])
show "measure lebesgue (\<Union>{cbox a b}) \<le> measure lebesgue S + e"
using assms by (simp add: sum_content.box_empty_imp [OF True])
qed (use assms \<open>cbox a b \<noteq> {}\<close> in auto)
qed
next
case False
let ?\<mu> = "measure lebesgue"
have "S \<inter> cbox a b \<in> lmeasurable"
using \<open>S \<in> lmeasurable\<close> by blast
then have indS_int: "(indicator S has_integral (?\<mu> S)) (cbox a b)"
by (metis integral_indicator \<open>S \<subseteq> cbox a b\<close> has_integral_integrable_integral inf.orderE integrable_on_indicator)
with \<open>e > 0\<close> obtain \<gamma> where "gauge \<gamma>" and \<gamma>:
"\<And>\<D>. \<lbrakk>\<D> tagged_division_of (cbox a b); \<gamma> fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x,K)\<in>\<D>. content(K) *\<^sub>R indicator S x) - ?\<mu> S) < e"
by (force simp: has_integral)
have inteq: "integral (cbox a b) (indicat_real S) = integral UNIV (indicator S)"
using assms by (metis has_integral_iff indS_int lmeasure_integral_UNIV)
obtain \<D> where \<D>: "countable \<D>" "\<Union>\<D> \<subseteq> cbox a b"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
and djointish: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
and covered: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> \<gamma> x"
and close: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v\<bullet>i - u\<bullet>i = (b\<bullet>i - a\<bullet>i)/2^n"
and covers: "S \<subseteq> \<Union>\<D>"
using covering_lemma [of S a b \<gamma>] \<open>gauge \<gamma>\<close> \<open>box a b \<noteq> {}\<close> assms by force
show ?thesis
proof
show "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
by (meson Sup_le_iff \<D>(2) cbox interior_empty)
have negl_int: "negligible(K \<inter> L)" if "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L" for K L
proof -
have "interior K \<inter> interior L = {}"
using djointish pairwiseD that by fastforce
moreover obtain u v x y where "K = cbox u v" "L = cbox x y"
using cbox \<open>K \<in> \<D>\<close> \<open>L \<in> \<D>\<close> by blast
ultimately show ?thesis
by (simp add: Int_interval box_Int_box negligible_interval(1))
qed
have fincase: "\<Union>\<F> \<in> lmeasurable \<and> ?\<mu> (\<Union>\<F>) \<le> ?\<mu> S + e" if "finite \<F>" "\<F> \<subseteq> \<D>" for \<F>
proof -
obtain t where t: "\<And>K. K \<in> \<F> \<Longrightarrow> t K \<in> S \<inter> K \<and> K \<subseteq> \<gamma>(t K)"
using covered \<open>\<F> \<subseteq> \<D>\<close> subsetD by metis
have "\<forall>K \<in> \<F>. \<forall>L \<in> \<F>. K \<noteq> L \<longrightarrow> interior K \<inter> interior L = {}"
using that djointish by (simp add: pairwise_def) (metis subsetD)
with cbox that \<D> have \<F>div: "\<F> division_of (\<Union>\<F>)"
by (fastforce simp: division_of_def dest: cbox)
then have 1: "\<Union>\<F> \<in> lmeasurable"
by blast
have norme: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
\<Longrightarrow> norm ((\<Sum>(x,K)\<in>p. content K * indicator S x) - integral (cbox a b) (indicator S)) < e"
by (auto simp: lmeasure_integral_UNIV assms inteq dest: \<gamma>)
have "\<forall>x K y L. (x,K) \<in> (\<lambda>K. (t K,K)) ` \<F> \<and> (y,L) \<in> (\<lambda>K. (t K,K)) ` \<F> \<and> (x,K) \<noteq> (y,L) \<longrightarrow> interior K \<inter> interior L = {}"
using that djointish by (clarsimp simp: pairwise_def) (metis subsetD)
with that \<D> have tagged: "(\<lambda>K. (t K, K)) ` \<F> tagged_partial_division_of cbox a b"
by (auto simp: tagged_partial_division_of_def dest: t cbox)
have fine: "\<gamma> fine (\<lambda>K. (t K, K)) ` \<F>"
using t by (auto simp: fine_def)
have *: "y \<le> ?\<mu> S \<Longrightarrow> \<bar>x - y\<bar> \<le> e \<Longrightarrow> x \<le> ?\<mu> S + e" for x y
by arith
have "?\<mu> (\<Union>\<F>) \<le> ?\<mu> S + e"
proof (rule *)
have "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) = ?\<mu> (\<Union>C\<in>\<F>. C \<inter> S)"
apply (rule measure_negligible_finite_Union_image [OF \<open>finite \<F>\<close>, symmetric])
using \<F>div \<open>S \<in> lmeasurable\<close> apply blast
unfolding pairwise_def
by (metis inf.commute inf_sup_aci(3) negligible_Int subsetCE negl_int \<open>\<F> \<subseteq> \<D>\<close>)
also have "\<dots> = ?\<mu> (\<Union>\<F> \<inter> S)"
by simp
also have "\<dots> \<le> ?\<mu> S"
by (simp add: "1" \<open>S \<in> lmeasurable\<close> fmeasurableD measure_mono_fmeasurable sets.Int)
finally show "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) \<le> ?\<mu> S" .
next
have "?\<mu> (\<Union>\<F>) = sum ?\<mu> \<F>"
by (metis \<F>div content_division)
also have "\<dots> = (\<Sum>K\<in>\<F>. content K)"
using \<F>div by (force intro: sum.cong)
also have "\<dots> = (\<Sum>x\<in>\<F>. content x * indicator S (t x))"
using t by auto
finally have eq1: "?\<mu> (\<Union>\<F>) = (\<Sum>x\<in>\<F>. content x * indicator S (t x))" .
have eq2: "(\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S)) = (\<Sum>K\<in>\<F>. integral K (indicator S))"
apply (rule sum.cong [OF refl])
by (metis integral_indicator \<F>div \<open>S \<in> lmeasurable\<close> division_ofD(4) fmeasurable.Int inf.commute lmeasurable_cbox)
have "\<bar>\<Sum>(x,K)\<in>(\<lambda>K. (t K, K)) ` \<F>. content K * indicator S x - integral K (indicator S)\<bar> \<le> e"
using Henstock_lemma_part1 [of "indicator S::'a\<Rightarrow>real", OF _ \<open>e > 0\<close> \<open>gauge \<gamma>\<close> _ tagged fine]
indS_int norme by auto
then show "\<bar>?\<mu> (\<Union>\<F>) - (\<Sum>K\<in>\<F>. ?\<mu> (K \<inter> S))\<bar> \<le> e"
by (simp add: eq1 eq2 comm_monoid_add_class.sum.reindex inj_on_def sum_subtractf)
qed
with 1 show ?thesis by blast
qed
have "\<Union>\<D> \<in> lmeasurable \<and> ?\<mu> (\<Union>\<D>) \<le> ?\<mu> S + e"
proof (cases "finite \<D>")
case True
with fincase show ?thesis
by blast
next
case False
let ?T = "from_nat_into \<D>"
have T: "bij_betw ?T UNIV \<D>"
by (simp add: False \<D>(1) bij_betw_from_nat_into)
have TM: "\<And>n. ?T n \<in> lmeasurable"
by (metis False cbox finite.emptyI from_nat_into lmeasurable_cbox)
have TN: "\<And>m n. m \<noteq> n \<Longrightarrow> negligible (?T m \<inter> ?T n)"
by (simp add: False \<D>(1) from_nat_into infinite_imp_nonempty negl_int)
have TB: "(\<Sum>k\<le>n. ?\<mu> (?T k)) \<le> ?\<mu> S + e" for n
proof -
have "(\<Sum>k\<le>n. ?\<mu> (?T k)) = ?\<mu> (\<Union> (?T ` {..n}))"
by (simp add: pairwise_def TM TN measure_negligible_finite_Union_image)
also have "?\<mu> (\<Union> (?T ` {..n})) \<le> ?\<mu> S + e"
using fincase [of "?T ` {..n}"] T by (auto simp: bij_betw_def)
finally show ?thesis .
qed
have "\<Union>\<D> \<in> lmeasurable"
by (metis lmeasurable_compact T \<D>(2) bij_betw_def cbox compact_cbox countable_Un_Int(1) fmeasurableD fmeasurableI2 rangeI)
moreover
have "?\<mu> (\<Union>x. from_nat_into \<D> x) \<le> ?\<mu> S + e"
proof (rule measure_countable_Union_le [OF TM])
show "?\<mu> (\<Union>x\<le>n. from_nat_into \<D> x) \<le> ?\<mu> S + e" for n
by (metis (mono_tags, lifting) False fincase finite.emptyI finite_atMost finite_imageI from_nat_into imageE subsetI)
qed
ultimately show ?thesis by (metis T bij_betw_def)
qed
then show "\<Union>\<D> \<in> lmeasurable" "measure lebesgue (\<Union>\<D>) \<le> ?\<mu> S + e" by blast+
qed (use \<D> cbox djointish close covers in auto)
qed
subsection\<open>Transformation of measure by linear maps\<close>
lemma measurable_linear_image_interval:
"linear f \<Longrightarrow> f ` (cbox a b) \<in> lmeasurable"
by (metis bounded_linear_image linear_linear bounded_cbox closure_bounded_linear_image closure_cbox compact_closure lmeasurable_compact)
proposition measure_linear_sufficient:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "linear f" and S: "S \<in> lmeasurable"
and im: "\<And>a b. measure lebesgue (f ` (cbox a b)) = m * measure lebesgue (cbox a b)"
shows "f ` S \<in> lmeasurable \<and> m * measure lebesgue S = measure lebesgue (f ` S)"
using le_less_linear [of 0 m]
proof
assume "m < 0"
then show ?thesis
using im [of 0 One] by auto
next
assume "m \<ge> 0"
let ?\<mu> = "measure lebesgue"
show ?thesis
proof (cases "inj f")
case False
then have "?\<mu> (f ` S) = 0"
using \<open>linear f\<close> negligible_imp_measure0 negligible_linear_singular_image by blast
then have "m * ?\<mu> (cbox 0 (One)) = 0"
by (metis False \<open>linear f\<close> cbox_borel content_unit im measure_completion negligible_imp_measure0 negligible_linear_singular_image sets_lborel)
then show ?thesis
using \<open>linear f\<close> negligible_linear_singular_image negligible_imp_measure0 False
by (auto simp: lmeasurable_iff_has_integral negligible_UNIV)
next
case True
then obtain h where "linear h" and hf: "\<And>x. h (f x) = x" and fh: "\<And>x. f (h x) = x"
using \<open>linear f\<close> linear_injective_isomorphism by blast
have fBS: "(f ` S) \<in> lmeasurable \<and> m * ?\<mu> S = ?\<mu> (f ` S)"
if "bounded S" "S \<in> lmeasurable" for S
proof -
obtain a b where "S \<subseteq> cbox a b"
using \<open>bounded S\<close> bounded_subset_cbox_symmetric by metis
have fUD: "(f ` \<Union>\<D>) \<in> lmeasurable \<and> ?\<mu> (f ` \<Union>\<D>) = (m * ?\<mu> (\<Union>\<D>))"
if "countable \<D>"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
and intint: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
for \<D>
proof -
have conv: "\<And>K. K \<in> \<D> \<Longrightarrow> convex K"
using cbox convex_box(1) by blast
have neg: "negligible (g ` K \<inter> g ` L)" if "linear g" "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L"
for K L and g :: "'n\<Rightarrow>'n"
proof (cases "inj g")
case True
have "negligible (frontier(g ` K \<inter> g ` L) \<union> interior(g ` K \<inter> g ` L))"
proof (rule negligible_Un)
show "negligible (frontier (g ` K \<inter> g ` L))"
by (simp add: negligible_convex_frontier convex_Int conv convex_linear_image that)
next
have "\<forall>p N. pairwise p N = (\<forall>Na. (Na::'n set) \<in> N \<longrightarrow> (\<forall>Nb. Nb \<in> N \<and> Na \<noteq> Nb \<longrightarrow> p Na Nb))"
by (metis pairwise_def)
then have "interior K \<inter> interior L = {}"
using intint that(2) that(3) that(4) by presburger
then show "negligible (interior (g ` K \<inter> g ` L))"
by (metis True empty_imp_negligible image_Int image_empty interior_Int interior_injective_linear_image that(1))
qed
moreover have "g ` K \<inter> g ` L \<subseteq> frontier (g ` K \<inter> g ` L) \<union> interior (g ` K \<inter> g ` L)"
apply (auto simp: frontier_def)
using closure_subset contra_subsetD by fastforce+
ultimately show ?thesis
by (rule negligible_subset)
next
case False
then show ?thesis
by (simp add: negligible_Int negligible_linear_singular_image \<open>linear g\<close>)
qed
have negf: "negligible ((f ` K) \<inter> (f ` L))"
and negid: "negligible (K \<inter> L)" if "K \<in> \<D>" "L \<in> \<D>" "K \<noteq> L" for K L
using neg [OF \<open>linear f\<close>] neg [OF linear_id] that by auto
show ?thesis
proof (cases "finite \<D>")
case True
then have "?\<mu> (\<Union>x\<in>\<D>. f ` x) = (\<Sum>x\<in>\<D>. ?\<mu> (f ` x))"
using \<open>linear f\<close> cbox measurable_linear_image_interval negf
by (blast intro: measure_negligible_finite_Union_image [unfolded pairwise_def])
also have "\<dots> = (\<Sum>k\<in>\<D>. m * ?\<mu> k)"
by (metis (no_types, lifting) cbox im sum.cong)
also have "\<dots> = m * ?\<mu> (\<Union>\<D>)"
unfolding sum_distrib_left [symmetric]
by (metis True cbox lmeasurable_cbox measure_negligible_finite_Union [unfolded pairwise_def] negid)
finally show ?thesis
by (metis True \<open>linear f\<close> cbox image_Union fmeasurable.finite_UN measurable_linear_image_interval)
next
case False
with \<open>countable \<D>\<close> obtain X :: "nat \<Rightarrow> 'n set" where S: "bij_betw X UNIV \<D>"
using bij_betw_from_nat_into by blast
then have eq: "(\<Union>\<D>) = (\<Union>n. X n)" "(f ` \<Union>\<D>) = (\<Union>n. f ` X n)"
by (auto simp: bij_betw_def)
have meas: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<in> lmeasurable"
using cbox by blast
with S have 1: "\<And>n. X n \<in> lmeasurable"
by (auto simp: bij_betw_def)
have 2: "pairwise (\<lambda>m n. negligible (X m \<inter> X n)) UNIV"
using S unfolding bij_betw_def pairwise_def by (metis injD negid range_eqI)
have "bounded (\<Union>\<D>)"
by (meson Sup_least bounded_cbox bounded_subset cbox)
then have 3: "bounded (\<Union>n. X n)"
using S unfolding bij_betw_def by blast
have "(\<Union>n. X n) \<in> lmeasurable"
by (rule measurable_countable_negligible_Union_bounded [OF 1 2 3])
with S have f1: "\<And>n. f ` (X n) \<in> lmeasurable"
unfolding bij_betw_def by (metis assms(1) cbox measurable_linear_image_interval rangeI)
have f2: "pairwise (\<lambda>m n. negligible (f ` (X m) \<inter> f ` (X n))) UNIV"
using S unfolding bij_betw_def pairwise_def by (metis injD negf rangeI)
have "bounded (\<Union>\<D>)"
by (meson Sup_least bounded_cbox bounded_subset cbox)
then have f3: "bounded (\<Union>n. f ` X n)"
using S unfolding bij_betw_def
by (metis bounded_linear_image linear_linear assms(1) image_Union range_composition)
have "(\<lambda>n. ?\<mu> (X n)) sums ?\<mu> (\<Union>n. X n)"
by (rule measure_countable_negligible_Union_bounded [OF 1 2 3])
have meq: "?\<mu> (\<Union>n. f ` X n) = m * ?\<mu> (\<Union>(X ` UNIV))"
proof (rule sums_unique2 [OF measure_countable_negligible_Union_bounded [OF f1 f2 f3]])
have m: "\<And>n. ?\<mu> (f ` X n) = (m * ?\<mu> (X n))"
using S unfolding bij_betw_def by (metis cbox im rangeI)
show "(\<lambda>n. ?\<mu> (f ` X n)) sums (m * ?\<mu> (\<Union>(X ` UNIV)))"
unfolding m
using measure_countable_negligible_Union_bounded [OF 1 2 3] sums_mult by blast
qed
show ?thesis
using measurable_countable_negligible_Union_bounded [OF f1 f2 f3] meq
by (auto simp: eq [symmetric])
qed
qed
show ?thesis
unfolding completion.fmeasurable_measure_inner_outer_le
proof (intro conjI allI impI)
fix e :: real
assume "e > 0"
have 1: "cbox a b - S \<in> lmeasurable"
by (simp add: fmeasurable.Diff that)
have 2: "0 < e / (1 + \<bar>m\<bar>)"
using \<open>e > 0\<close> by (simp add: divide_simps abs_add_one_gt_zero)
obtain \<D>
where "countable \<D>"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
and intdisj: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
and DD: "cbox a b - S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable"
and le: "?\<mu> (\<Union>\<D>) \<le> ?\<mu> (cbox a b - S) + e/(1 + \<bar>m\<bar>)"
by (rule measurable_outer_intervals_bounded [of "cbox a b - S" a b "e/(1 + \<bar>m\<bar>)"]; use 1 2 pairwise_def in force)
have meq: "?\<mu> (cbox a b - S) = ?\<mu> (cbox a b) - ?\<mu> S"
by (simp add: measurable_measure_Diff \<open>S \<subseteq> cbox a b\<close> fmeasurableD that(2))
show "\<exists>T \<in> lmeasurable. T \<subseteq> f ` S \<and> m * ?\<mu> S - e \<le> ?\<mu> T"
proof (intro bexI conjI)
show "f ` (cbox a b) - f ` (\<Union>\<D>) \<subseteq> f ` S"
using \<open>cbox a b - S \<subseteq> \<Union>\<D>\<close> by force
have "m * ?\<mu> S - e \<le> m * (?\<mu> S - e / (1 + \<bar>m\<bar>))"
using \<open>m \<ge> 0\<close> \<open>e > 0\<close> by (simp add: field_simps)
also have "\<dots> \<le> ?\<mu> (f ` cbox a b) - ?\<mu> (f ` (\<Union>\<D>))"
using le \<open>m \<ge> 0\<close> \<open>e > 0\<close>
apply (simp add: im fUD [OF \<open>countable \<D>\<close> cbox intdisj] right_diff_distrib [symmetric])
apply (rule mult_left_mono; simp add: algebra_simps meq)
done
also have "\<dots> = ?\<mu> (f ` cbox a b - f ` \<Union>\<D>)"
apply (rule measurable_measure_Diff [symmetric])
apply (simp add: assms(1) measurable_linear_image_interval)
apply (simp add: \<open>countable \<D>\<close> cbox fUD fmeasurableD intdisj)
apply (simp add: Sup_le_iff cbox image_mono)
done
finally show "m * ?\<mu> S - e \<le> ?\<mu> (f ` cbox a b - f ` \<Union>\<D>)" .
show "f ` cbox a b - f ` \<Union>\<D> \<in> lmeasurable"
by (simp add: fUD \<open>countable \<D>\<close> \<open>linear f\<close> cbox fmeasurable.Diff intdisj measurable_linear_image_interval)
qed
next
fix e :: real
assume "e > 0"
have em: "0 < e / (1 + \<bar>m\<bar>)"
using \<open>e > 0\<close> by (simp add: divide_simps abs_add_one_gt_zero)
obtain \<D>
where "countable \<D>"
and cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b \<and> K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
and intdisj: "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
and DD: "S \<subseteq> \<Union>\<D>" "\<Union>\<D> \<in> lmeasurable"
and le: "?\<mu> (\<Union>\<D>) \<le> ?\<mu> S + e/(1 + \<bar>m\<bar>)"
by (rule measurable_outer_intervals_bounded [of S a b "e/(1 + \<bar>m\<bar>)"]; use \<open>S \<in> lmeasurable\<close> \<open>S \<subseteq> cbox a b\<close> em in force)
show "\<exists>U \<in> lmeasurable. f ` S \<subseteq> U \<and> ?\<mu> U \<le> m * ?\<mu> S + e"
proof (intro bexI conjI)
show "f ` S \<subseteq> f ` (\<Union>\<D>)"
by (simp add: DD(1) image_mono)
have "?\<mu> (f ` \<Union>\<D>) \<le> m * (?\<mu> S + e / (1 + \<bar>m\<bar>))"
using \<open>m \<ge> 0\<close> le mult_left_mono
by (auto simp: fUD \<open>countable \<D>\<close> \<open>linear f\<close> cbox fmeasurable.Diff intdisj measurable_linear_image_interval)
also have "\<dots> \<le> m * ?\<mu> S + e"
using \<open>m \<ge> 0\<close> \<open>e > 0\<close> by (simp add: fUD [OF \<open>countable \<D>\<close> cbox intdisj] field_simps)
finally show "?\<mu> (f ` \<Union>\<D>) \<le> m * ?\<mu> S + e" .
show "f ` \<Union>\<D> \<in> lmeasurable"
by (simp add: \<open>countable \<D>\<close> cbox fUD intdisj)
qed
qed
qed
show ?thesis
unfolding has_measure_limit_iff
proof (intro allI impI)
fix e :: real
assume "e > 0"
obtain B where "B > 0" and B:
"\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> \<bar>?\<mu> (S \<inter> cbox a b) - ?\<mu> S\<bar> < e / (1 + \<bar>m\<bar>)"
using has_measure_limit [OF S] \<open>e > 0\<close> by (metis abs_add_one_gt_zero zero_less_divide_iff)
obtain c d::'n where cd: "ball 0 B \<subseteq> cbox c d"
by (metis bounded_subset_cbox_symmetric bounded_ball)
with B have less: "\<bar>?\<mu> (S \<inter> cbox c d) - ?\<mu> S\<bar> < e / (1 + \<bar>m\<bar>)" .
obtain D where "D > 0" and D: "cbox c d \<subseteq> ball 0 D"
by (metis bounded_cbox bounded_subset_ballD)
obtain C where "C > 0" and C: "\<And>x. norm (f x) \<le> C * norm x"
using linear_bounded_pos \<open>linear f\<close> by blast
have "f ` S \<inter> cbox a b \<in> lmeasurable \<and>
\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e"
if "ball 0 (D*C) \<subseteq> cbox a b" for a b
proof -
have "bounded (S \<inter> h ` cbox a b)"
by (simp add: bounded_linear_image linear_linear \<open>linear h\<close> bounded_Int)
moreover have Shab: "S \<inter> h ` cbox a b \<in> lmeasurable"
by (simp add: S \<open>linear h\<close> fmeasurable.Int measurable_linear_image_interval)
moreover have fim: "f ` (S \<inter> h ` (cbox a b)) = (f ` S) \<inter> cbox a b"
by (auto simp: hf rev_image_eqI fh)
ultimately have 1: "(f ` S) \<inter> cbox a b \<in> lmeasurable"
and 2: "m * ?\<mu> (S \<inter> h ` cbox a b) = ?\<mu> ((f ` S) \<inter> cbox a b)"
using fBS [of "S \<inter> (h ` (cbox a b))"] by auto
have *: "\<lbrakk>\<bar>z - m\<bar> < e; z \<le> w; w \<le> m\<rbrakk> \<Longrightarrow> \<bar>w - m\<bar> \<le> e"
for w z m and e::real by auto
have meas_adiff: "\<bar>?\<mu> (S \<inter> h ` cbox a b) - ?\<mu> S\<bar> \<le> e / (1 + \<bar>m\<bar>)"
proof (rule * [OF less])
show "?\<mu> (S \<inter> cbox c d) \<le> ?\<mu> (S \<inter> h ` cbox a b)"
proof (rule measure_mono_fmeasurable [OF _ _ Shab])
have "f ` ball 0 D \<subseteq> ball 0 (C * D)"
using C \<open>C > 0\<close>
apply (clarsimp simp: algebra_simps)
by (meson le_less_trans linordered_comm_semiring_strict_class.comm_mult_strict_left_mono)
then have "f ` ball 0 D \<subseteq> cbox a b"
by (metis mult.commute order_trans that)
have "ball 0 D \<subseteq> h ` cbox a b"
by (metis \<open>f ` ball 0 D \<subseteq> cbox a b\<close> hf image_subset_iff subsetI)
then show "S \<inter> cbox c d \<subseteq> S \<inter> h ` cbox a b"
using D by blast
next
show "S \<inter> cbox c d \<in> sets lebesgue"
using S fmeasurable_cbox by blast
qed
next
show "?\<mu> (S \<inter> h ` cbox a b) \<le> ?\<mu> S"
by (simp add: S Shab fmeasurableD measure_mono_fmeasurable)
qed
have "\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> \<le> m * e / (1 + \<bar>m\<bar>)"
proof -
have mm: "\<bar>m\<bar> = m"
by (simp add: \<open>0 \<le> m\<close>)
then have "\<bar>?\<mu> S - ?\<mu> (S \<inter> h ` cbox a b)\<bar> * m \<le> e / (1 + m) * m"
by (metis (no_types) \<open>0 \<le> m\<close> meas_adiff abs_minus_commute mult_right_mono)
moreover have "\<forall>r. \<bar>r * m\<bar> = m * \<bar>r\<bar>"
by (metis \<open>0 \<le> m\<close> abs_mult_pos mult.commute)
ultimately show ?thesis
apply (simp add: 2 [symmetric])
by (metis (no_types) abs_minus_commute mult.commute right_diff_distrib' mm)
qed
also have "\<dots> < e"
using \<open>e > 0\<close> by (auto simp: divide_simps)
finally have "\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e" .
with 1 show ?thesis by auto
qed
then show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
f ` S \<inter> cbox a b \<in> lmeasurable \<and>
\<bar>?\<mu> (f ` S \<inter> cbox a b) - m * ?\<mu> S\<bar> < e"
using \<open>C>0\<close> \<open>D>0\<close> by (metis mult_zero_left real_mult_less_iff1)
qed
qed
qed
subsection\<open>Lemmas about absolute integrability\<close>
text\<open>FIXME Redundant!\<close>
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)
text\<open>FIXME Redundant!\<close>
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] set_integrable_def)
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 by induction auto
lemma absolutely_integrable_integrable_bound:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes le: "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> g x" and f: "f integrable_on S" and g: "g integrable_on S"
shows "f absolutely_integrable_on S"
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound)
have "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
then show "integrable lebesgue (\<lambda>x. indicat_real S x *\<^sub>R g x)"
by (simp add: set_integrable_def)
show "(\<lambda>x. indicat_real S x *\<^sub>R f x) \<in> borel_measurable lebesgue"
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
qed (use le in \<open>force intro!: always_eventually split: split_indicator\<close>)
lemma absolutely_integrable_continuous:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "continuous_on (cbox a b) f \<Longrightarrow> f absolutely_integrable_on cbox a b"
using absolutely_integrable_integrable_bound
by (simp add: absolutely_integrable_on_def continuous_on_norm integrable_continuous)
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"
using absolutely_integrable_componentwise_iff by blast
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
using assms by blast
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 by (simp add: absolutely_integrable_on_def o_def)
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"
using abs_absolutely_integrableI_1 f integrable_component by blast
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
lemma integrable_on_1_iff:
fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
shows "f integrable_on S \<longleftrightarrow> (\<lambda>x. f x $ 1) integrable_on S"
by (auto simp: integrable_componentwise_iff [of f] Basis_vec_def cart_eq_inner_axis)
lemma integral_on_1_eq:
fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
shows "integral S f = vec (integral S (\<lambda>x. f x $ 1))"
by (cases "f integrable_on S") (simp_all add: integrable_on_1_iff vec_eq_iff not_integrable_integral)
lemma absolutely_integrable_on_1_iff:
fixes f :: "'a::euclidean_space \<Rightarrow> real^1"
shows "f absolutely_integrable_on S \<longleftrightarrow> (\<lambda>x. f x $ 1) absolutely_integrable_on S"
unfolding absolutely_integrable_on_def
by (auto simp: integrable_on_1_iff norm_real)
lemma absolutely_integrable_absolutely_integrable_lbound:
fixes f :: "'m::euclidean_space \<Rightarrow> real"
assumes f: "f integrable_on S" and g: "g absolutely_integrable_on S"
and *: "\<And>x. x \<in> S \<Longrightarrow> g x \<le> f x"
shows "f absolutely_integrable_on S"
by (rule absolutely_integrable_component_lbound [OF g f]) (simp add: *)
lemma absolutely_integrable_absolutely_integrable_ubound:
fixes f :: "'m::euclidean_space \<Rightarrow> real"
assumes fg: "f integrable_on S" "g absolutely_integrable_on S"
and *: "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
shows "f absolutely_integrable_on S"
by (rule absolutely_integrable_component_ubound [OF fg]) (simp add: *)
lemma has_integral_vec1_I_cbox:
fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
assumes "(f has_integral y) (cbox a b)"
shows "((f \<circ> vec) has_integral y) {a$1..b$1}"
proof -
have "((\<lambda>x. f(vec x)) has_integral (1 / 1) *\<^sub>R y) ((\<lambda>x. x $ 1) ` cbox a b)"
proof (rule has_integral_twiddle)
show "\<exists>w z::real^1. vec ` cbox u v = cbox w z"
"content (vec ` cbox u v :: (real^1) set) = 1 * content (cbox u v)" for u v
unfolding vec_cbox_1_eq
by (auto simp: content_cbox_if_cart interval_eq_empty_cart)
show "\<exists>w z. (\<lambda>x. x $ 1) ` cbox u v = cbox w z" for u v :: "real^1"
using vec_nth_cbox_1_eq by blast
qed (auto simp: continuous_vec assms)
then show ?thesis
by (simp add: o_def)
qed
lemma has_integral_vec1_I:
fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
assumes "(f has_integral y) S"
shows "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
proof -
have *: "\<exists>z. ((\<lambda>x. if x \<in> (\<lambda>x. x $ 1) ` S then (f \<circ> vec) x else 0) has_integral z) {a..b} \<and> norm (z - y) < e"
if int: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow>
(\<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)"
and B: "ball 0 B \<subseteq> {a..b}" for e B a b
proof -
have [simp]: "(\<exists>y\<in>S. x = y $ 1) \<longleftrightarrow> vec x \<in> S" for x
by force
have B': "ball (0::real^1) B \<subseteq> cbox (vec a) (vec b)"
using B by (simp add: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box norm_real subset_iff)
show ?thesis
using int [OF B'] by (auto simp: image_iff o_def cong: if_cong dest!: has_integral_vec1_I_cbox)
qed
show ?thesis
using assms
apply (subst has_integral_alt)
apply (subst (asm) has_integral_alt)
apply (simp add: has_integral_vec1_I_cbox split: if_split_asm)
apply (metis vector_one_nth)
apply (erule all_forward imp_forward asm_rl ex_forward conj_forward)+
apply (blast intro!: *)
done
qed
lemma has_integral_vec1_nth_cbox:
fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
assumes "(f has_integral y) {a..b}"
shows "((\<lambda>x::real^1. f(x$1)) has_integral y) (cbox (vec a) (vec b))"
proof -
have "((\<lambda>x::real^1. f(x$1)) has_integral (1 / 1) *\<^sub>R y) (vec ` cbox a b)"
proof (rule has_integral_twiddle)
show "\<exists>w z::real. (\<lambda>x. x $ 1) ` cbox u v = cbox w z"
"content ((\<lambda>x. x $ 1) ` cbox u v) = 1 * content (cbox u v)" for u v::"real^1"
unfolding vec_cbox_1_eq by (auto simp: content_cbox_if_cart interval_eq_empty_cart)
show "\<exists>w z::real^1. vec ` cbox u v = cbox w z" for u v :: "real"
using vec_cbox_1_eq by auto
qed (auto simp: continuous_vec assms)
then show ?thesis
using vec_cbox_1_eq by auto
qed
lemma has_integral_vec1_D_cbox:
fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
assumes "((f \<circ> vec) has_integral y) {a$1..b$1}"
shows "(f has_integral y) (cbox a b)"
by (metis (mono_tags, lifting) assms comp_apply has_integral_eq has_integral_vec1_nth_cbox vector_one_nth)
lemma has_integral_vec1_D:
fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
assumes "((f \<circ> vec) has_integral y) ((\<lambda>x. x $ 1) ` S)"
shows "(f has_integral y) S"
proof -
have *: "\<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e"
if int: "\<And>a b. ball 0 B \<subseteq> {a..b} \<Longrightarrow>
(\<exists>z. ((\<lambda>x. if x \<in> (\<lambda>x. x $ 1) ` S then (f \<circ> vec) x else 0) has_integral z) {a..b} \<and> norm (z - y) < e)"
and B: "ball 0 B \<subseteq> cbox a b" for e B and a b::"real^1"
proof -
have B': "ball 0 B \<subseteq> {a$1..b$1}"
using B
apply (simp add: subset_iff Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
apply (metis (full_types) norm_real vec_component)
done
have eq: "(\<lambda>x. if vec x \<in> S then f (vec x) else 0) = (\<lambda>x. if x \<in> S then f x else 0) \<circ> vec"
by force
have [simp]: "(\<exists>y\<in>S. x = y $ 1) \<longleftrightarrow> vec x \<in> S" for x
by force
show ?thesis
using int [OF B'] by (auto simp: image_iff eq cong: if_cong dest!: has_integral_vec1_D_cbox)
qed
show ?thesis
using assms
apply (subst has_integral_alt)
apply (subst (asm) has_integral_alt)
apply (simp add: has_integral_vec1_D_cbox eq_cbox split: if_split_asm, blast)
apply (intro conjI impI)
apply (metis vector_one_nth)
apply (erule thin_rl)
apply (erule all_forward imp_forward asm_rl ex_forward conj_forward)+
using * apply blast
done
qed
lemma integral_vec1_eq:
fixes f :: "real^1 \<Rightarrow> 'a::real_normed_vector"
shows "integral S f = integral ((\<lambda>x. x $ 1) ` S) (f \<circ> vec)"
using has_integral_vec1_I [of f] has_integral_vec1_D [of f]
by (metis has_integral_iff not_integrable_integral)
lemma absolutely_integrable_drop:
fixes f :: "real^1 \<Rightarrow> 'b::euclidean_space"
shows "f absolutely_integrable_on S \<longleftrightarrow> (f \<circ> vec) absolutely_integrable_on (\<lambda>x. x $ 1) ` S"
unfolding absolutely_integrable_on_def integrable_on_def
proof safe
fix y r
assume "(f has_integral y) S" "((\<lambda>x. norm (f x)) has_integral r) S"
then show "\<exists>y. (f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
"\<exists>y. ((\<lambda>x. norm ((f \<circ> vec) x)) has_integral y) ((\<lambda>x. x $ 1) ` S)"
by (force simp: o_def dest!: has_integral_vec1_I)+
next
fix y :: "'b" and r :: "real"
assume "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
"((\<lambda>x. norm ((f \<circ> vec) x)) has_integral r) ((\<lambda>x. x $ 1) ` S)"
then show "\<exists>y. (f has_integral y) S" "\<exists>y. ((\<lambda>x. norm (f x)) has_integral y) S"
by (force simp: o_def intro: has_integral_vec1_D)+
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 x. x \<in> S \<Longrightarrow> 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
unfolding set_borel_measurable_def
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
then have 1: "set_borel_measurable lebesgue S g"
unfolding set_borel_measurable_def
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 unfolding set_borel_measurable_def set_integrable_def
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)"
unfolding set_borel_measurable_def set_lebesgue_integral_def
using 1 2 3 4 unfolding set_borel_measurable_def set_lebesgue_integral_def set_integrable_def
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 (metis assms(2-4) dominated_convergence(1) has_integral_integral int_f)
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 (metis assms(2) assms(3) assms(4) dominated_convergence(2) int_f)
qed
ultimately show ?thesis
by simp
qed
lemma dominated_convergence_integrable_1:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
assumes f: "\<And>k. f k absolutely_integrable_on S"
and h: "h integrable_on S"
and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
shows "g integrable_on S"
proof -
have habs: "h absolutely_integrable_on S"
using h normg nonnegative_absolutely_integrable_1 norm_ge_zero order_trans by blast
let ?f = "\<lambda>n x. (min (max (- h x) (f n x)) (h x))"
have h0: "h x \<ge> 0" if "x \<in> S" for x
using normg that by force
have leh: "norm (?f k x) \<le> h x" if "x \<in> S" for k x
using h0 that by force
have limf: "(\<lambda>k. ?f k x) \<longlonglongrightarrow> g x" if "x \<in> S" for x
proof -
have "\<And>e y. \<bar>f y x - g x\<bar> < e \<Longrightarrow> \<bar>min (max (- h x) (f y x)) (h x) - g x\<bar> < e"
using h0 [OF that] normg [OF that] by simp
then show ?thesis
using lim [OF that] by (auto simp add: tendsto_iff dist_norm elim!: eventually_mono)
qed
show ?thesis
proof (rule dominated_convergence [of ?f S h g])
have "(\<lambda>x. - h x) absolutely_integrable_on S"
using habs unfolding set_integrable_def by auto
then show "?f k integrable_on S" for k
by (intro set_lebesgue_integral_eq_integral absolutely_integrable_min_1 absolutely_integrable_max_1 f habs)
qed (use assms leh limf in auto)
qed
lemma dominated_convergence_integrable:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "\<And>k. f k absolutely_integrable_on S"
and h: "h integrable_on S"
and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
shows "g integrable_on S"
using f
unfolding integrable_componentwise_iff [of g] absolutely_integrable_componentwise_iff [where f = "f k" for k]
proof clarify
fix b :: "'m"
assume fb [rule_format]: "\<And>k. \<forall>b\<in>Basis. (\<lambda>x. f k x \<bullet> b) absolutely_integrable_on S" and b: "b \<in> Basis"
show "(\<lambda>x. g x \<bullet> b) integrable_on S"
proof (rule dominated_convergence_integrable_1 [OF fb h])
fix x
assume "x \<in> S"
show "norm (g x \<bullet> b) \<le> h x"
using norm_nth_le \<open>x \<in> S\<close> b normg order.trans by blast
show "(\<lambda>k. f k x \<bullet> b) \<longlonglongrightarrow> g x \<bullet> b"
using \<open>x \<in> S\<close> b lim tendsto_componentwise_iff by fastforce
qed (use b in auto)
qed
lemma dominated_convergence_absolutely_integrable:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "\<And>k. f k absolutely_integrable_on S"
and h: "h integrable_on S"
and normg: "\<And>x. x \<in> S \<Longrightarrow> norm(g x) \<le> (h x)"
and lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
shows "g absolutely_integrable_on S"
proof -
have "g integrable_on S"
by (rule dominated_convergence_integrable [OF assms])
with assms show ?thesis
by (blast intro: absolutely_integrable_integrable_bound [where g=h])
qed
proposition integral_countable_UN:
fixes f :: "real^'m \<Rightarrow> real^'n"
assumes f: "f absolutely_integrable_on (\<Union>(range s))"
and s: "\<And>m. s m \<in> sets lebesgue"
shows "\<And>n. f absolutely_integrable_on (\<Union>m\<le>n. s m)"
and "(\<lambda>n. integral (\<Union>m\<le>n. s m) f) \<longlonglongrightarrow> integral (\<Union>(s ` UNIV)) f" (is "?F \<longlonglongrightarrow> ?I")
proof -
show fU: "f absolutely_integrable_on (\<Union>m\<le>n. s m)" for n
using assms by (blast intro: set_integrable_subset [OF f])
have fint: "f integrable_on (\<Union> (range s))"
using absolutely_integrable_on_def f by blast
let ?h = "\<lambda>x. if x \<in> \<Union>(s ` UNIV) then norm(f x) else 0"
have "(\<lambda>n. integral UNIV (\<lambda>x. if x \<in> (\<Union>m\<le>n. s m) then f x else 0))
\<longlonglongrightarrow> integral UNIV (\<lambda>x. if x \<in> \<Union>(s ` UNIV) then f x else 0)"
proof (rule dominated_convergence)
show "(\<lambda>x. if x \<in> (\<Union>m\<le>n. s m) then f x else 0) integrable_on UNIV" for n
unfolding integrable_restrict_UNIV
using fU absolutely_integrable_on_def by blast
show "(\<lambda>x. if x \<in> \<Union>(s ` UNIV) then norm(f x) else 0) integrable_on UNIV"
by (metis (no_types) absolutely_integrable_on_def f integrable_restrict_UNIV)
show "\<forall>x\<in>UNIV.
(\<lambda>n. if x \<in> (\<Union>m\<le>n. s m) then f x else 0)
\<longlonglongrightarrow> (if x \<in> \<Union>(s ` UNIV) then f x else 0)"
by (force intro: Lim_eventually eventually_sequentiallyI)
qed auto
then show "?F \<longlonglongrightarrow> ?I"
by (simp only: integral_restrict_UNIV)
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: eventually_at_top_linorder)
have "(SUP i. ?f i x) = ?fR x" for x
proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
obtain n where "x - a < real n"
using reals_Archimedean2[of "x - a"] ..
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. ?f i x) \<partial>lborel)"
by simp
also have "\<dots> = (SUP i. (\<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. 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)"
unfolding set_lebesgue_integral_def
proof (rule integral_nonneg_eq_0_iff_AE)
show "integrable lebesgue (\<lambda>x. indicat_real (closure S) x *\<^sub>R f x)"
by (metis af set_integrable_def)
qed (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
subsection\<open>Various common equivalent forms of function measurability\<close>
lemma indicator_sum_eq:
fixes m::real and f :: "'a \<Rightarrow> real"
assumes "\<bar>m\<bar> \<le> 2 ^ (2*n)" "m/2^n \<le> f x" "f x < (m+1)/2^n" "m \<in> \<int>"
shows "(\<Sum>k::real | k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n).
k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) = m/2^n"
(is "sum ?f ?S = _")
proof -
have "sum ?f ?S = sum (\<lambda>k. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x) {m}"
proof (rule comm_monoid_add_class.sum.mono_neutral_right)
show "finite ?S"
by (rule finite_abs_int_segment)
show "{m} \<subseteq> {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
using assms by auto
show "\<forall>i\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)} - {m}. ?f i = 0"
using assms by (auto simp: indicator_def Ints_def abs_le_iff divide_simps)
qed
also have "\<dots> = m/2^n"
using assms by (auto simp: indicator_def not_less)
finally show ?thesis .
qed
lemma measurable_on_sf_limit_lemma1:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
assumes "\<And>a b. {x \<in> S. a \<le> f x \<and> f x < b} \<in> sets (lebesgue_on S)"
obtains g where "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
"\<And>n. finite(range (g n))"
"\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x"
proof
show "(\<lambda>x. sum (\<lambda>k::real. k/2^n * indicator {y. k/2^n \<le> f y \<and> f y < (k+1)/2^n} x)
{k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}) \<in> borel_measurable (lebesgue_on S)"
(is "?g \<in> _") for n
proof -
have "\<And>k. \<lbrakk>k \<in> \<int>; \<bar>k\<bar> \<le> 2 ^ (2*n)\<rbrakk>
\<Longrightarrow> Measurable.pred (lebesgue_on S) (\<lambda>x. k / (2^n) \<le> f x \<and> f x < (k+1) / (2^n))"
using assms by (force simp: pred_def space_restrict_space)
then show ?thesis
by (simp add: field_class.field_divide_inverse)
qed
show "finite (range (?g n))" for n
proof -
have "range (?g n) \<subseteq> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
proof clarify
fix x
show "?g n x \<in> (\<lambda>k. k/2^n) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)}"
proof (cases "\<exists>k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2 ^ (2*n) \<and> k/2^n \<le> (f x) \<and> (f x) < (k+1)/2^n")
case True
then show ?thesis
apply clarify
by (subst indicator_sum_eq) auto
next
case False
then have "?g n x = 0" by auto
then show ?thesis by force
qed
qed
moreover have "finite ((\<lambda>k::real. (k/2^n)) ` {k \<in> \<int>. \<bar>k\<bar> \<le> 2 ^ (2*n)})"
by (simp add: finite_abs_int_segment)
ultimately show ?thesis
using finite_subset by blast
qed
show "(\<lambda>n. ?g n x) \<longlonglongrightarrow> f x" for x
proof (rule LIMSEQ_I)
fix e::real
assume "e > 0"
obtain N1 where N1: "\<bar>f x\<bar> < 2 ^ N1"
using real_arch_pow by fastforce
obtain N2 where N2: "(1/2) ^ N2 < e"
using real_arch_pow_inv \<open>e > 0\<close> by force
have "norm (?g n x - f x) < e" if n: "n \<ge> max N1 N2" for n
proof -
define m where "m \<equiv> floor(2^n * (f x))"
have "1 \<le> \<bar>2^n\<bar> * e"
using n N2 \<open>e > 0\<close> less_eq_real_def less_le_trans by (fastforce simp add: divide_simps)
then have *: "\<lbrakk>x \<le> y; y < x + 1\<rbrakk> \<Longrightarrow> abs(x - y) < \<bar>2^n\<bar> * e" for x y::real
by linarith
have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> = \<bar>2^n * (m/2^n - f x)\<bar>"
by (simp add: abs_mult)
also have "\<dots> = \<bar>real_of_int \<lfloor>2^n * f x\<rfloor> - f x * 2^n\<bar>"
by (simp add: algebra_simps m_def)
also have "\<dots> < \<bar>2^n\<bar> * e"
by (rule *; simp add: mult.commute)
finally have "\<bar>2^n\<bar> * \<bar>m/2^n - f x\<bar> < \<bar>2^n\<bar> * e" .
then have me: "\<bar>m/2^n - f x\<bar> < e"
by simp
have "\<bar>real_of_int m\<bar> \<le> 2 ^ (2*n)"
proof (cases "f x < 0")
case True
then have "-\<lfloor>f x\<rfloor> \<le> \<lfloor>(2::real) ^ N1\<rfloor>"
using N1 le_floor_iff minus_le_iff by fastforce
with n True have "\<bar>real_of_int\<lfloor>f x\<rfloor>\<bar> \<le> 2 ^ N1"
by linarith
also have "\<dots> \<le> 2^n"
using n by (simp add: m_def)
finally have "\<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n \<le> 2^n * 2^n"
by simp
moreover
have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
proof -
have "\<bar>real_of_int \<lfloor>2^n * f x\<rfloor>\<bar> = - (real_of_int \<lfloor>2^n * f x\<rfloor>)"
using True by (simp add: abs_if mult_less_0_iff)
also have "\<dots> \<le> - (real_of_int (\<lfloor>(2::real) ^ n\<rfloor> * \<lfloor>f x\<rfloor>))"
using le_mult_floor_Ints [of "(2::real)^n"] by simp
also have "\<dots> \<le> \<bar>real_of_int \<lfloor>f x\<rfloor>\<bar> * 2^n"
using True
by simp
finally show ?thesis .
qed
ultimately show ?thesis
by (metis (no_types, hide_lams) m_def order_trans power2_eq_square power_even_eq)
next
case False
with n N1 have "f x \<le> 2^n"
by (simp add: not_less) (meson less_eq_real_def one_le_numeral order_trans power_increasing)
moreover have "0 \<le> m"
using False m_def by force
ultimately show ?thesis
by (metis abs_of_nonneg floor_mono le_floor_iff m_def of_int_0_le_iff power2_eq_square power_mult real_mult_le_cancel_iff1 zero_less_numeral mult.commute zero_less_power)
qed
then have "?g n x = m/2^n"
by (rule indicator_sum_eq) (auto simp: m_def mult.commute divide_simps)
then have "norm (?g n x - f x) = norm (m/2^n - f x)"
by simp
also have "\<dots> < e"
by (simp add: me)
finally show ?thesis .
qed
then show "\<exists>no. \<forall>n\<ge>no. norm (?g n x - f x) < e"
by blast
qed
qed
lemma borel_measurable_vimage_halfspace_component_lt:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_less])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_simple_function_limit:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
(\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
proof -
have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
(\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x)"
if f: "\<And>a i. i \<in> Basis \<Longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S)"
proof -
have "\<exists>g. (\<forall>n. (g n) \<in> borel_measurable (lebesgue_on S)) \<and>
(\<forall>n. finite(image (g n) UNIV)) \<and>
(\<forall>x. ((\<lambda>n. g n x) \<longlonglongrightarrow> f x \<bullet> i))" if "i \<in> Basis" for i
proof (rule measurable_on_sf_limit_lemma1 [of S "\<lambda>x. f x \<bullet> i"])
show "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} \<in> sets (lebesgue_on S)" for a b
proof -
have "{x \<in> S. a \<le> f x \<bullet> i \<and> f x \<bullet> i < b} = {x \<in> S. f x \<bullet> i < b} - {x \<in> S. a > f x \<bullet> i}"
by auto
also have "\<dots> \<in> sets (lebesgue_on S)"
using f that by blast
finally show ?thesis .
qed
qed blast
then obtain g where g:
"\<And>i n. i \<in> Basis \<Longrightarrow> g i n \<in> borel_measurable (lebesgue_on S)"
"\<And>i n. i \<in> Basis \<Longrightarrow> finite(range (g i n))"
"\<And>i x. i \<in> Basis \<Longrightarrow> ((\<lambda>n. g i n x) \<longlonglongrightarrow> f x \<bullet> i)"
by metis
show ?thesis
proof (intro conjI allI exI)
show "(\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> borel_measurable (lebesgue_on S)" for n
by (intro borel_measurable_sum borel_measurable_scaleR) (auto intro: g)
show "finite (range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i))" for n
proof -
have "range (\<lambda>x. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<subseteq> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` PiE Basis (\<lambda>i. range (g i n))"
proof clarify
fix x
show "(\<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<in> (\<lambda>h. \<Sum>i\<in>Basis. h i *\<^sub>R i) ` (\<Pi>\<^sub>E i\<in>Basis. range (g i n))"
by (rule_tac x="\<lambda>i\<in>Basis. g i n x" in image_eqI) auto
qed
moreover have "finite(PiE Basis (\<lambda>i. range (g i n)))"
by (simp add: g finite_PiE)
ultimately show ?thesis
by (metis (mono_tags, lifting) finite_surj)
qed
show "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> f x" for x
proof -
have "(\<lambda>n. \<Sum>i\<in>Basis. g i n x *\<^sub>R i) \<longlonglongrightarrow> (\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i)"
by (auto intro!: tendsto_sum tendsto_scaleR g)
moreover have "(\<Sum>i\<in>Basis. (f x \<bullet> i) *\<^sub>R i) = f x"
using euclidean_representation by blast
ultimately show ?thesis
by metis
qed
qed
qed
moreover have "f \<in> borel_measurable (lebesgue_on S)"
if meas_g: "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"
and fin: "\<And>n. finite (range (g n))"
and to_f: "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" for g
by (rule borel_measurable_LIMSEQ_metric [OF meas_g to_f])
ultimately show ?thesis
using borel_measurable_vimage_halfspace_component_lt by blast
qed
lemma borel_measurable_vimage_halfspace_component_ge:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<ge> a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_ge])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_vimage_halfspace_component_gt:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i > a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_greater])
apply (fastforce simp add: space_restrict_space)
done
lemma borel_measurable_vimage_halfspace_component_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<le> a} \<in> sets (lebesgue_on S))"
apply (rule trans [OF borel_measurable_iff_halfspace_le])
apply (fastforce simp add: space_restrict_space)
done
lemma
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows borel_measurable_vimage_open_interval:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S))" (is ?thesis1)
and borel_measurable_vimage_open:
"f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))" (is ?thesis2)
proof -
have "{x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" if "f \<in> borel_measurable (lebesgue_on S)" for a b
proof -
have "S = S \<inter> space lebesgue"
by simp
then have "S \<inter> (f -` box a b) \<in> sets (lebesgue_on S)"
by (metis (no_types) box_borel in_borel_measurable_borel inf_sup_aci(1) space_restrict_space that)
then show ?thesis
by (simp add: Collect_conj_eq vimage_def)
qed
moreover
have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
if T: "\<And>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S)" "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = T"
using open_countable_Union_open_box that \<open>open T\<close> by metis
then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
by blast
have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
using that T \<D> by blast
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
qed
moreover
have eq: "{x \<in> S. f x \<bullet> i < a} = {x \<in> S. f x \<in> {y. y \<bullet> i < a}}" for i a
by auto
have "f \<in> borel_measurable (lebesgue_on S)"
if "\<And>T. open T \<Longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)"
by (metis (no_types) eq borel_measurable_vimage_halfspace_component_lt open_halfspace_component_lt that)
ultimately show "?thesis1" "?thesis2"
by blast+
qed
lemma borel_measurable_vimage_closed:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. closed T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof -
have eq: "{x \<in> S. f x \<in> T} = S - {x \<in> S. f x \<in> (- T)}" for T
by auto
show ?thesis
apply (simp add: borel_measurable_vimage_open, safe)
apply (simp_all (no_asm) add: eq)
apply (intro sets.Diff sets_lebesgue_on_refl, force simp: closed_open)
apply (intro sets.Diff sets_lebesgue_on_refl, force simp: open_closed)
done
qed
lemma borel_measurable_vimage_closed_interval:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using borel_measurable_vimage_closed by blast
next
assume RHS: ?rhs
have "{x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S)" if "open T" for T
proof -
obtain \<D> where "countable \<D>" and \<D>: "\<D> \<subseteq> Pow T" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = T"
using open_countable_Union_open_cbox that \<open>open T\<close> by metis
then have eq: "{x \<in> S. f x \<in> T} = (\<Union>U \<in> \<D>. {x \<in> S. f x \<in> U})"
by blast
have "{x \<in> S. f x \<in> U} \<in> sets (lebesgue_on S)" if "U \<in> \<D>" for U
using that \<D> by (metis RHS)
then show ?thesis
by (auto simp: eq intro: Sigma_Algebra.sets.countable_UN' [OF \<open>countable \<D>\<close>])
qed
then show ?lhs
by (simp add: borel_measurable_vimage_open)
qed
lemma borel_measurable_UNIV_eq: "borel_measurable (lebesgue_on UNIV) = borel_measurable lebesgue"
by auto
lemma borel_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
(is "?lhs = ?rhs")
proof
assume f: ?lhs
then show ?rhs
using measurable_sets [OF f]
by (simp add: Collect_conj_eq inf_sup_aci(1) space_restrict_space vimage_def)
qed (simp add: borel_measurable_vimage_open_interval)
lemma lebesgue_measurable_vimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f \<in> borel_measurable lebesgue" "T \<in> sets borel"
shows "{x. f x \<in> T} \<in> sets lebesgue"
using assms borel_measurable_vimage_borel [of f UNIV] by auto
lemma borel_measurable_If_I:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f \<in> borel_measurable (lebesgue_on S)" and S: "S \<in> sets lebesgue"
shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
proof -
have eq: "{x. x \<notin> S} \<union> {x. f x \<in> Y} = {x. x \<notin> S} \<union> {x. f x \<in> Y} \<inter> S" for Y
by blast
show ?thesis
using f S
apply (simp add: vimage_def in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (simp only: Collect_conj_eq Collect_disj_eq imp_conv_disj eq)
apply (auto simp: Compl_eq [symmetric] Compl_in_sets_lebesgue sets_restrict_space_iff)
done
qed
lemma borel_measurable_If_D:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
shows "f \<in> borel_measurable (lebesgue_on S)"
using assms
apply (simp add: in_borel_measurable_borel Ball_def)
apply (elim all_forward imp_forward asm_rl)
apply (force simp: space_restrict_space sets_restrict_space image_iff intro: rev_bexI)
done
lemma borel_measurable_UNIV:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue"
shows "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue \<longleftrightarrow> f \<in> borel_measurable (lebesgue_on S)"
using assms borel_measurable_If_D borel_measurable_If_I by blast
lemma borel_measurable_lebesgue_preimage_borel:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
shows "f \<in> borel_measurable lebesgue \<longleftrightarrow>
(\<forall>T. T \<in> sets borel \<longrightarrow> {x. f x \<in> T} \<in> sets lebesgue)"
apply (intro iffI allI impI lebesgue_measurable_vimage_borel)
apply (auto simp: in_borel_measurable_borel vimage_def)
done
subsection\<open>More results on integrability\<close>
lemma integrable_on_all_intervals_UNIV:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
assumes intf: "\<And>a b. f integrable_on cbox a b"
and normf: "\<And>x. norm(f x) \<le> g x" and g: "g integrable_on UNIV"
shows "f integrable_on UNIV"
proof -
have intg: "(\<forall>a b. g integrable_on cbox a b)"
and gle_e: "\<forall>e>0. \<exists>B>0. \<forall>a b c d.
ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
\<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>
< e"
using g
by (auto simp: integrable_alt_subset [of _ UNIV] intf)
have le: "norm (integral (cbox a b) f - integral (cbox c d) f) \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
if "cbox a b \<subseteq> cbox c d" for a b c d
proof -
have "norm (integral (cbox a b) f - integral (cbox c d) f) = norm (integral (cbox c d - cbox a b) f)"
using intf that by (simp add: norm_minus_commute integral_setdiff)
also have "\<dots> \<le> integral (cbox c d - cbox a b) g"
proof (rule integral_norm_bound_integral [OF _ _ normf])
show "f integrable_on cbox c d - cbox a b" "g integrable_on cbox c d - cbox a b"
by (meson integrable_integral integrable_setdiff intf intg negligible_setdiff that)+
qed
also have "\<dots> = integral (cbox c d) g - integral (cbox a b) g"
using intg that by (simp add: integral_setdiff)
also have "\<dots> \<le> \<bar>integral (cbox a b) g - integral (cbox c d) g\<bar>"
by simp
finally show ?thesis .
qed
show ?thesis
using gle_e
apply (simp add: integrable_alt_subset [of _ UNIV] intf)
apply (erule imp_forward all_forward ex_forward asm_rl)+
by (meson not_less order_trans le)
qed
lemma integrable_on_all_intervals_integrable_bound:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
assumes intf: "\<And>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b"
and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and g: "g integrable_on S"
shows "f integrable_on S"
using integrable_on_all_intervals_UNIV [OF intf, of "(\<lambda>x. if x \<in> S then g x else 0)"]
by (simp add: g integrable_restrict_UNIV normf)
lemma measurable_bounded_lemma:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f \<in> borel_measurable lebesgue" and g: "g integrable_on cbox a b"
and normf: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm(f x) \<le> g x"
shows "f integrable_on cbox a b"
proof -
have "g absolutely_integrable_on cbox a b"
by (metis (full_types) add_increasing g le_add_same_cancel1 nonnegative_absolutely_integrable_1 norm_ge_zero normf)
then have "integrable (lebesgue_on (cbox a b)) g"
by (simp add: integrable_restrict_space set_integrable_def)
then have "integrable (lebesgue_on (cbox a b)) f"
proof (rule Bochner_Integration.integrable_bound)
show "AE x in lebesgue_on (cbox a b). norm (f x) \<le> norm (g x)"
by (rule AE_I2) (auto intro: normf order_trans)
qed (simp add: f measurable_restrict_space1)
then show ?thesis
by (simp add: integrable_on_lebesgue_on)
qed
proposition measurable_bounded_by_integrable_imp_integrable:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f \<in> borel_measurable (lebesgue_on S)" and g: "g integrable_on S"
and normf: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> g x" and S: "S \<in> sets lebesgue"
shows "f integrable_on S"
proof (rule integrable_on_all_intervals_integrable_bound [OF _ normf g])
show "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b" for a b
proof (rule measurable_bounded_lemma)
show "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable lebesgue"
by (simp add: S borel_measurable_UNIV f)
show "(\<lambda>x. if x \<in> S then g x else 0) integrable_on cbox a b"
by (simp add: g integrable_altD(1))
show "norm (if x \<in> S then f x else 0) \<le> (if x \<in> S then g x else 0)" for x
using normf by simp
qed
qed
subsection\<open> Relation between Borel measurability and integrability.\<close>
lemma integrable_imp_measurable_weak:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "S \<in> sets lebesgue" "f integrable_on S"
shows "f \<in> borel_measurable (lebesgue_on S)"
by (metis (mono_tags, lifting) assms has_integral_implies_lebesgue_measurable borel_measurable_restrict_space_iff integrable_on_def sets.Int_space_eq2)
lemma integrable_imp_measurable:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f integrable_on S"
shows "f \<in> borel_measurable (lebesgue_on S)"
proof -
have "(UNIV::'a set) \<in> sets lborel"
by simp
then show ?thesis
using assms borel_measurable_If_D borel_measurable_UNIV_eq integrable_imp_measurable_weak integrable_restrict_UNIV by blast
qed
proposition negligible_differentiable_vimage:
fixes f :: "'a \<Rightarrow> 'a::euclidean_space"
assumes "negligible T"
and f': "\<And>x. x \<in> S \<Longrightarrow> inj(f' x)"
and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
shows "negligible {x \<in> S. f x \<in> T}"
proof -
define U where
"U \<equiv> \<lambda>n::nat. {x \<in> S. \<forall>y. y \<in> S \<and> norm(y - x) < 1/n
\<longrightarrow> norm(y - x) \<le> n * norm(f y - f x)}"
have "negligible {x \<in> U n. f x \<in> T}" if "n > 0" for n
proof (subst locally_negligible_alt, clarify)
fix a
assume a: "a \<in> U n" and fa: "f a \<in> T"
define V where "V \<equiv> {x. x \<in> U n \<and> f x \<in> T} \<inter> ball a (1 / n / 2)"
show "\<exists>V. openin (subtopology euclidean {x \<in> U n. f x \<in> T}) V \<and> a \<in> V \<and> negligible V"
proof (intro exI conjI)
have noxy: "norm(x - y) \<le> n * norm(f x - f y)" if "x \<in> V" "y \<in> V" for x y
using that unfolding U_def V_def mem_Collect_eq Int_iff mem_ball dist_norm
by (meson norm_triangle_half_r)
then have "inj_on f V"
by (force simp: inj_on_def)
then obtain g where g: "\<And>x. x \<in> V \<Longrightarrow> g(f x) = x"
by (metis inv_into_f_f)
have "\<exists>T' B. open T' \<and> f x \<in> T' \<and>
(\<forall>y\<in>f ` V \<inter> T \<inter> T'. norm (g y - g (f x)) \<le> B * norm (y - f x))"
if "f x \<in> T" "x \<in> V" for x
apply (rule_tac x = "ball (f x) 1" in exI)
using that noxy by (force simp: g)
then have "negligible (g ` (f ` V \<inter> T))"
by (force simp: \<open>negligible T\<close> negligible_Int intro!: negligible_locally_Lipschitz_image)
moreover have "V \<subseteq> g ` (f ` V \<inter> T)"
by (force simp: g image_iff V_def)
ultimately show "negligible V"
by (rule negligible_subset)
qed (use a fa V_def that in auto)
qed
with negligible_countable_Union have "negligible (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
by auto
moreover have "{x \<in> S. f x \<in> T} \<subseteq> (\<Union>n \<in> {0<..}. {x. x \<in> U n \<and> f x \<in> T})"
proof clarsimp
fix x
assume "x \<in> S" and "f x \<in> T"
then obtain inj: "inj(f' x)" and der: "(f has_derivative f' x) (at x within S)"
using assms by metis
moreover have "linear(f' x)"
and eps: "\<And>\<epsilon>. \<epsilon> > 0 \<Longrightarrow> \<exists>\<delta>>0. \<forall>y\<in>S. norm (y - x) < \<delta> \<longrightarrow>
norm (f y - f x - f' x (y - x)) \<le> \<epsilon> * norm (y - x)"
using der by (auto simp: has_derivative_within_alt linear_linear)
ultimately obtain g where "linear g" and g: "g \<circ> f' x = id"
using linear_injective_left_inverse by metis
then obtain B where "B > 0" and B: "\<And>z. B * norm z \<le> norm(f' x z)"
using linear_invertible_bounded_below_pos \<open>linear (f' x)\<close> by blast
then obtain i where "i \<noteq> 0" and i: "1 / real i < B"
by (metis (full_types) inverse_eq_divide real_arch_invD)
then obtain \<delta> where "\<delta> > 0"
and \<delta>: "\<And>y. \<lbrakk>y\<in>S; norm (y - x) < \<delta>\<rbrakk> \<Longrightarrow>
norm (f y - f x - f' x (y - x)) \<le> (B - 1 / real i) * norm (y - x)"
using eps [of "B - 1/i"] by auto
then obtain j where "j \<noteq> 0" and j: "inverse (real j) < \<delta>"
using real_arch_inverse by blast
have "norm (y - x)/(max i j) \<le> norm (f y - f x)"
if "y \<in> S" and less: "norm (y - x) < 1 / (max i j)" for y
proof -
have "1 / real (max i j) < \<delta>"
using j \<open>j \<noteq> 0\<close> \<open>0 < \<delta>\<close>
by (auto simp: divide_simps max_mult_distrib_left of_nat_max)
then have "norm (y - x) < \<delta>"
using less by linarith
with \<delta> \<open>y \<in> S\<close> have le: "norm (f y - f x - f' x (y - x)) \<le> B * norm (y - x) - norm (y - x)/i"
by (auto simp: algebra_simps)
have *: "\<lbrakk>norm(f - f' - y) \<le> b - c; b \<le> norm y; d \<le> c\<rbrakk> \<Longrightarrow> d \<le> norm(f - f')"
for b c d and y f f'::'a
using norm_triangle_ineq3 [of "f - f'" y] by simp
show ?thesis
apply (rule * [OF le B])
using \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> by (simp add: divide_simps max_mult_distrib_left of_nat_max less_max_iff_disj)
qed
with \<open>x \<in> S\<close> \<open>i \<noteq> 0\<close> \<open>j \<noteq> 0\<close> show "\<exists>n\<in>{0<..}. x \<in> U n"
by (rule_tac x="max i j" in bexI) (auto simp: field_simps U_def less_max_iff_disj)
qed
ultimately show ?thesis
by (rule negligible_subset)
qed
lemma absolutely_integrable_Un:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes S: "f absolutely_integrable_on S" and T: "f absolutely_integrable_on T"
shows "f absolutely_integrable_on (S \<union> T)"
proof -
have [simp]: "{x. (if x \<in> A then f x else 0) \<noteq> 0} = {x \<in> A. f x \<noteq> 0}" for A
by auto
let ?ST = "{x \<in> S. f x \<noteq> 0} \<inter> {x \<in> T. f x \<noteq> 0}"
have "?ST \<in> sets lebesgue"
proof (rule Sigma_Algebra.sets.Int)
have "f integrable_on S"
using S absolutely_integrable_on_def by blast
then have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on UNIV"
by (simp add: integrable_restrict_UNIV)
then have borel: "(\<lambda>x. if x \<in> S then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
using integrable_imp_measurable borel_measurable_UNIV_eq by blast
then show "{x \<in> S. f x \<noteq> 0} \<in> sets lebesgue"
unfolding borel_measurable_vimage_open
by (rule allE [where x = "-{0}"]) auto
next
have "f integrable_on T"
using T absolutely_integrable_on_def by blast
then have "(\<lambda>x. if x \<in> T then f x else 0) integrable_on UNIV"
by (simp add: integrable_restrict_UNIV)
then have borel: "(\<lambda>x. if x \<in> T then f x else 0) \<in> borel_measurable (lebesgue_on UNIV)"
using integrable_imp_measurable borel_measurable_UNIV_eq by blast
then show "{x \<in> T. f x \<noteq> 0} \<in> sets lebesgue"
unfolding borel_measurable_vimage_open
by (rule allE [where x = "-{0}"]) auto
qed
then have "f absolutely_integrable_on ?ST"
by (rule set_integrable_subset [OF S]) auto
then have Int: "(\<lambda>x. if x \<in> ?ST then f x else 0) absolutely_integrable_on UNIV"
using absolutely_integrable_restrict_UNIV by blast
have "(\<lambda>x. if x \<in> S then f x else 0) absolutely_integrable_on UNIV"
"(\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
using S T absolutely_integrable_restrict_UNIV by blast+
then have "(\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) absolutely_integrable_on UNIV"
by (rule absolutely_integrable_add)
then have "(\<lambda>x. ((if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) - (if x \<in> ?ST then f x else 0)) absolutely_integrable_on UNIV"
using Int by (rule absolutely_integrable_diff)
then have "(\<lambda>x. if x \<in> S \<union> T then f x else 0) absolutely_integrable_on UNIV"
by (rule absolutely_integrable_spike) (auto intro: empty_imp_negligible)
then show ?thesis
unfolding absolutely_integrable_restrict_UNIV .
qed
lemma uniform_limit_set_lebesgue_integral_at_top:
fixes f :: "'a \<Rightarrow> real \<Rightarrow> 'b::{banach, second_countable_topology}"
and g :: "real \<Rightarrow> real"
assumes bound: "\<And>x y. x \<in> A \<Longrightarrow> y \<ge> a \<Longrightarrow> norm (f x y) \<le> g y"
assumes integrable: "set_integrable M {a..} g"
assumes measurable: "\<And>x. x \<in> A \<Longrightarrow> set_borel_measurable M {a..} (f x)"
assumes "sets borel \<subseteq> sets M"
shows "uniform_limit A (\<lambda>b x. LINT y:{a..b}|M. f x y) (\<lambda>x. LINT y:{a..}|M. f x y) at_top"
proof (cases "A = {}")
case False
then obtain x where x: "x \<in> A" by auto
have g_nonneg: "g y \<ge> 0" if "y \<ge> a" for y
proof -
have "0 \<le> norm (f x y)" by simp
also have "\<dots> \<le> g y" using bound[OF x that] by simp
finally show ?thesis .
qed
have integrable': "set_integrable M {a..} (\<lambda>y. f x y)" if "x \<in> A" for x
unfolding set_integrable_def
proof (rule Bochner_Integration.integrable_bound)
show "integrable M (\<lambda>x. indicator {a..} x * g x)"
using integrable by (simp add: set_integrable_def)
show "(\<lambda>y. indicat_real {a..} y *\<^sub>R f x y) \<in> borel_measurable M" using measurable[OF that]
by (simp add: set_borel_measurable_def)
show "AE y in M. norm (indicat_real {a..} y *\<^sub>R f x y) \<le> norm (indicat_real {a..} y * g y)"
using bound[OF that] by (intro AE_I2) (auto simp: indicator_def g_nonneg)
qed
show ?thesis
proof (rule uniform_limitI)
fix e :: real assume e: "e > 0"
have sets [intro]: "A \<in> sets M" if "A \<in> sets borel" for A
using that assms by blast
have "((\<lambda>b. LINT y:{a..b}|M. g y) \<longlongrightarrow> (LINT y:{a..}|M. g y)) at_top"
by (intro tendsto_set_lebesgue_integral_at_top assms sets) auto
with e obtain b0 :: real where b0: "\<forall>b\<ge>b0. \<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
by (auto simp: tendsto_iff eventually_at_top_linorder dist_real_def abs_minus_commute)
define b where "b = max a b0"
have "a \<le> b" by (simp add: b_def)
from b0 have "\<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
by (auto simp: b_def)
also have "{a..} = {a..b} \<union> {b<..}" by (auto simp: b_def)
also have "\<bar>(LINT y:\<dots>|M. g y) - (LINT y:{a..b}|M. g y)\<bar> = \<bar>(LINT y:{b<..}|M. g y)\<bar>"
using \<open>a \<le> b\<close> by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable])
also have "(LINT y:{b<..}|M. g y) \<ge> 0"
using g_nonneg \<open>a \<le> b\<close> unfolding set_lebesgue_integral_def
by (intro Bochner_Integration.integral_nonneg) (auto simp: indicator_def)
hence "\<bar>(LINT y:{b<..}|M. g y)\<bar> = (LINT y:{b<..}|M. g y)" by simp
finally have less: "(LINT y:{b<..}|M. g y) < e" .
have "eventually (\<lambda>b. b \<ge> b0) at_top" by (rule eventually_ge_at_top)
moreover have "eventually (\<lambda>b. b \<ge> a) at_top" by (rule eventually_ge_at_top)
ultimately show "eventually (\<lambda>b. \<forall>x\<in>A.
dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) < e) at_top"
proof eventually_elim
case (elim b)
show ?case
proof
fix x assume x: "x \<in> A"
have "dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) =
norm ((LINT y:{a..}|M. f x y) - (LINT y:{a..b}|M. f x y))"
by (simp add: dist_norm norm_minus_commute)
also have "{a..} = {a..b} \<union> {b<..}" using elim by auto
also have "(LINT y:\<dots>|M. f x y) - (LINT y:{a..b}|M. f x y) = (LINT y:{b<..}|M. f x y)"
using elim x
by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable'])
also have "norm \<dots> \<le> (LINT y:{b<..}|M. norm (f x y))" using elim x
by (intro set_integral_norm_bound set_integrable_subset[OF integrable']) auto
also have "\<dots> \<le> (LINT y:{b<..}|M. g y)" using elim x bound g_nonneg
by (intro set_integral_mono set_integrable_norm set_integrable_subset[OF integrable']
set_integrable_subset[OF integrable]) auto
also have "(LINT y:{b<..}|M. g y) \<ge> 0"
using g_nonneg \<open>a \<le> b\<close> unfolding set_lebesgue_integral_def
by (intro Bochner_Integration.integral_nonneg) (auto simp: indicator_def)
hence "(LINT y:{b<..}|M. g y) = \<bar>(LINT y:{b<..}|M. g y)\<bar>" by simp
also have "\<dots> = \<bar>(LINT y:{a..b} \<union> {b<..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar>"
using elim by (subst set_integral_Un) (auto intro!: set_integrable_subset[OF integrable])
also have "{a..b} \<union> {b<..} = {a..}" using elim by auto
also have "\<bar>(LINT y:{a..}|M. g y) - (LINT y:{a..b}|M. g y)\<bar> < e"
using b0 elim by blast
finally show "dist (LINT y:{a..b}|M. f x y) (LINT y:{a..}|M. f x y) < e" .
qed
qed
qed
qed auto
subsubsection\<open>Differentiability of inverse function (most basic form)\<close>
proposition has_derivative_inverse_within:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
assumes der_f: "(f has_derivative f') (at a within S)"
and cont_g: "continuous (at (f a) within f ` S) g"
and "a \<in> S" "linear g'" and id: "g' \<circ> f' = id"
and gf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
shows "(g has_derivative g') (at (f a) within f ` S)"
proof -
have [simp]: "g' (f' x) = x" for x
by (simp add: local.id pointfree_idE)
have "bounded_linear f'"
and f': "\<And>e. e>0 \<Longrightarrow> \<exists>d>0. \<forall>y\<in>S. norm (y - a) < d \<longrightarrow>
norm (f y - f a - f' (y - a)) \<le> e * norm (y - a)"
using der_f by (auto simp: has_derivative_within_alt)
obtain C where "C > 0" and C: "\<And>x. norm (g' x) \<le> C * norm x"
using linear_bounded_pos [OF \<open>linear g'\<close>] by metis
obtain B k where "B > 0" "k > 0"
and Bk: "\<And>x. \<lbrakk>x \<in> S; norm(f x - f a) < k\<rbrakk> \<Longrightarrow> norm(x - a) \<le> B * norm(f x - f a)"
proof -
obtain B where "B > 0" and B: "\<And>x. B * norm x \<le> norm (f' x)"
using linear_inj_bounded_below_pos [of f'] \<open>linear g'\<close> id der_f has_derivative_linear
linear_invertible_bounded_below_pos by blast
then obtain d where "d>0"
and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
norm (f y - f a - f' (y - a)) \<le> B / 2 * norm (y - a)"
using f' [of "B/2"] by auto
then obtain e where "e > 0"
and e: "\<And>x. \<lbrakk>x \<in> S; norm (f x - f a) < e\<rbrakk> \<Longrightarrow> norm (g (f x) - g (f a)) < d"
using cont_g by (auto simp: continuous_within_eps_delta dist_norm)
show thesis
proof
show "2/B > 0"
using \<open>B > 0\<close> by simp
show "norm (x - a) \<le> 2 / B * norm (f x - f a)"
if "x \<in> S" "norm (f x - f a) < e" for x
proof -
have xa: "norm (x - a) < d"
using e [OF that] gf by (simp add: \<open>a \<in> S\<close> that)
have *: "\<lbrakk>norm(y - f') \<le> B / 2 * norm x; B * norm x \<le> norm f'\<rbrakk>
\<Longrightarrow> norm y \<ge> B / 2 * norm x" for y f'::'b and x::'a
using norm_triangle_ineq3 [of y f'] by linarith
show ?thesis
using * [OF d [OF \<open>x \<in> S\<close> xa] B] \<open>B > 0\<close> by (simp add: field_simps)
qed
qed (use \<open>e > 0\<close> in auto)
qed
show ?thesis
unfolding has_derivative_within_alt
proof (intro conjI impI allI)
show "bounded_linear g'"
using \<open>linear g'\<close> by (simp add: linear_linear)
next
fix e :: "real"
assume "e > 0"
then obtain d where "d>0"
and d: "\<And>y. \<lbrakk>y \<in> S; norm (y - a) < d\<rbrakk> \<Longrightarrow>
norm (f y - f a - f' (y - a)) \<le> e / (B * C) * norm (y - a)"
using f' [of "e / (B * C)"] \<open>B > 0\<close> \<open>C > 0\<close> by auto
have "norm (x - a - g' (f x - f a)) \<le> e * norm (f x - f a)"
if "x \<in> S" and lt_k: "norm (f x - f a) < k" and lt_dB: "norm (f x - f a) < d/B" for x
proof -
have "norm (x - a) \<le> B * norm(f x - f a)"
using Bk lt_k \<open>x \<in> S\<close> by blast
also have "\<dots> < d"
by (metis \<open>0 < B\<close> lt_dB mult.commute pos_less_divide_eq)
finally have lt_d: "norm (x - a) < d" .
have "norm (x - a - g' (f x - f a)) \<le> norm(g'(f x - f a - (f' (x - a))))"
by (simp add: linear_diff [OF \<open>linear g'\<close>] norm_minus_commute)
also have "\<dots> \<le> C * norm (f x - f a - f' (x - a))"
using C by blast
also have "\<dots> \<le> e * norm (f x - f a)"
proof -
have "norm (f x - f a - f' (x - a)) \<le> e / (B * C) * norm (x - a)"
using d [OF \<open>x \<in> S\<close> lt_d] .
also have "\<dots> \<le> (norm (f x - f a) * e) / C"
using \<open>B > 0\<close> \<open>C > 0\<close> \<open>e > 0\<close> by (simp add: field_simps Bk lt_k \<open>x \<in> S\<close>)
finally show ?thesis
using \<open>C > 0\<close> by (simp add: field_simps)
qed
finally show ?thesis .
qed
then show "\<exists>d>0. \<forall>y\<in>f ` S.
norm (y - f a) < d \<longrightarrow>
norm (g y - g (f a) - g' (y - f a)) \<le> e * norm (y - f a)"
apply (rule_tac x="min k (d / B)" in exI)
using \<open>k > 0\<close> \<open>B > 0\<close> \<open>d > 0\<close> \<open>a \<in> S\<close> by (auto simp: gf)
qed
qed
end