section%important \<open>Continuity of the indefinite integral; improper integral theorem\<close>
theory "Improper_Integral"
imports Equivalence_Lebesgue_Henstock_Integration
begin
subsection%important \<open>Equiintegrability\<close>
text\<open>The definition here only really makes sense for an elementary set.
We just use compact intervals in applications below.\<close>
definition%important equiintegrable_on (infixr "equiintegrable'_on" 46)
where "F equiintegrable_on I \<equiv>
(\<forall>f \<in> F. f integrable_on I) \<and>
(\<forall>e > 0. \<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma> fine \<D>
\<longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < e))"
lemma equiintegrable_on_integrable:
"\<lbrakk>F equiintegrable_on I; f \<in> F\<rbrakk> \<Longrightarrow> f integrable_on I"
using equiintegrable_on_def by metis
lemma equiintegrable_on_sing [simp]:
"{f} equiintegrable_on cbox a b \<longleftrightarrow> f integrable_on cbox a b"
by (simp add: equiintegrable_on_def has_integral_integral has_integral integrable_on_def)
lemma equiintegrable_on_subset: "\<lbrakk>F equiintegrable_on I; G \<subseteq> F\<rbrakk> \<Longrightarrow> G equiintegrable_on I"
unfolding equiintegrable_on_def Ball_def
by (erule conj_forward imp_forward all_forward ex_forward | blast)+
lemma%important equiintegrable_on_Un:
assumes "F equiintegrable_on I" "G equiintegrable_on I"
shows "(F \<union> G) equiintegrable_on I"
unfolding equiintegrable_on_def
proof%unimportant (intro conjI impI allI)
show "\<forall>f\<in>F \<union> G. f integrable_on I"
using assms unfolding equiintegrable_on_def by blast
show "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>f \<D>. f \<in> F \<union> G \<and>
\<D> tagged_division_of I \<and> \<gamma> fine \<D> \<longrightarrow>
norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>)"
if "\<epsilon> > 0" for \<epsilon>
proof -
obtain \<gamma>1 where "gauge \<gamma>1"
and \<gamma>1: "\<And>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma>1 fine \<D>
\<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by auto
obtain \<gamma>2 where "gauge \<gamma>2"
and \<gamma>2: "\<And>f \<D>. f \<in> G \<and> \<D> tagged_division_of I \<and> \<gamma>2 fine \<D>
\<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by auto
have "gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
using \<open>gauge \<gamma>1\<close> \<open>gauge \<gamma>2\<close> by blast
moreover have "\<forall>f \<D>. f \<in> F \<union> G \<and> \<D> tagged_division_of I \<and> (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x) fine \<D> \<longrightarrow>
norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
using \<gamma>1 \<gamma>2 by (auto simp: fine_Int)
ultimately show ?thesis
by (intro exI conjI) assumption+
qed
qed
lemma equiintegrable_on_insert:
assumes "f integrable_on cbox a b" "F equiintegrable_on cbox a b"
shows "(insert f F) equiintegrable_on cbox a b"
by (metis assms equiintegrable_on_Un equiintegrable_on_sing insert_is_Un)
text\<open> Basic combining theorems for the interval of integration.\<close>
lemma equiintegrable_on_null [simp]:
"content(cbox a b) = 0 \<Longrightarrow> F equiintegrable_on cbox a b"
apply (auto simp: equiintegrable_on_def)
by (metis gauge_trivial norm_eq_zero sum_content_null)
text\<open> Main limit theorem for an equiintegrable sequence.\<close>
theorem%important equiintegrable_limit:
fixes g :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
assumes feq: "range f equiintegrable_on cbox a b"
and to_g: "\<And>x. x \<in> cbox a b \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x"
shows "g integrable_on cbox a b \<and> (\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> integral (cbox a b) g"
proof%unimportant -
have "Cauchy (\<lambda>n. integral(cbox a b) (f n))"
proof (clarsimp simp add: Cauchy_def)
fix e::real
assume "0 < e"
then have e3: "0 < e/3"
by simp
then obtain \<gamma> where "gauge \<gamma>"
and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
\<Longrightarrow> norm((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) < e/3"
using feq unfolding equiintegrable_on_def
by (meson image_eqI iso_tuple_UNIV_I)
obtain \<D> where \<D>: "\<D> tagged_division_of (cbox a b)" and "\<gamma> fine \<D>" "finite \<D>"
by (meson \<open>gauge \<gamma>\<close> fine_division_exists tagged_division_of_finite)
with \<gamma> have \<delta>T: "\<And>n. dist ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)) (integral (cbox a b) (f n)) < e/3"
by (force simp: dist_norm)
have "(\<lambda>n. \<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) \<longlonglongrightarrow> (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x)"
using \<D> to_g by (auto intro!: tendsto_sum tendsto_scaleR)
then have "Cauchy (\<lambda>n. \<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)"
by (meson convergent_eq_Cauchy)
with e3 obtain M where
M: "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> dist (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f m x) (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)
< e/3"
unfolding Cauchy_def by blast
have "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M;
dist (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f m x) (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) < e/3\<rbrakk>
\<Longrightarrow> dist (integral (cbox a b) (f m)) (integral (cbox a b) (f n)) < e"
by (metis \<delta>T dist_commute dist_triangle_third [OF _ _ \<delta>T])
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (integral (cbox a b) (f m)) (integral (cbox a b) (f n)) < e"
using M by auto
qed
then obtain L where L: "(\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> L"
by (meson convergent_eq_Cauchy)
have "(g has_integral L) (cbox a b)"
proof (clarsimp simp: has_integral)
fix e::real assume "0 < e"
then have e2: "0 < e/2"
by simp
then obtain \<gamma> where "gauge \<gamma>"
and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
\<Longrightarrow> norm((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) < e/2"
using feq unfolding equiintegrable_on_def
by (meson image_eqI iso_tuple_UNIV_I)
moreover
have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) < e"
if "\<D> tagged_division_of cbox a b" "\<gamma> fine \<D>" for \<D>
proof -
have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) \<le> e/2"
proof (rule Lim_norm_ubound)
show "(\<lambda>n. (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) \<longlonglongrightarrow> (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L"
using to_g that L
by (intro tendsto_diff tendsto_sum) (auto simp: tag_in_interval tendsto_scaleR)
show "\<forall>\<^sub>F n in sequentially.
norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) \<le> e/2"
by (intro eventuallyI less_imp_le \<gamma> that)
qed auto
with \<open>0 < e\<close> show ?thesis
by linarith
qed
ultimately
show "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) < e)"
by meson
qed
with L show ?thesis
by (simp add: \<open>(\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> L\<close> has_integral_integrable_integral)
qed
lemma%important equiintegrable_reflect:
assumes "F equiintegrable_on cbox a b"
shows "(\<lambda>f. f \<circ> uminus) ` F equiintegrable_on cbox (-b) (-a)"
proof%unimportant -
have "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>f \<D>. f \<in> (\<lambda>f. f \<circ> uminus) ` F \<and> \<D> tagged_division_of cbox (- b) (- a) \<and> \<gamma> fine \<D> \<longrightarrow>
norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox (- b) (- a)) f) < e)"
if "gauge \<gamma>" and
\<gamma>: "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow>
norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox a b) f) < e" for e \<gamma>
proof (intro exI, safe)
show "gauge (\<lambda>x. uminus ` \<gamma> (-x))"
by (metis \<open>gauge \<gamma>\<close> gauge_reflect)
show "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R (f \<circ> uminus) x) - integral (cbox (- b) (- a)) (f \<circ> uminus)) < e"
if "f \<in> F" and tag: "\<D> tagged_division_of cbox (- b) (- a)"
and fine: "(\<lambda>x. uminus ` \<gamma> (- x)) fine \<D>" for f \<D>
proof -
have 1: "(\<lambda>(x,K). (- x, uminus ` K)) ` \<D> tagged_partial_division_of cbox a b"
if "\<D> tagged_partial_division_of cbox (- b) (- a)"
proof -
have "- y \<in> cbox a b"
if "\<And>x K. (x,K) \<in> \<D> \<Longrightarrow> x \<in> K \<and> K \<subseteq> cbox (- b) (- a) \<and> (\<exists>a b. K = cbox a b)"
"(x, Y) \<in> \<D>" "y \<in> Y" for x Y y
proof -
have "y \<in> uminus ` cbox a b"
using that by auto
then show "- y \<in> cbox a b"
by force
qed
with that show ?thesis
by (fastforce simp: tagged_partial_division_of_def interior_negations image_iff)
qed
have 2: "\<exists>K. (\<exists>x. (x,K) \<in> (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>) \<and> x \<in> K"
if "\<Union>{K. \<exists>x. (x,K) \<in> \<D>} = cbox (- b) (- a)" "x \<in> cbox a b" for x
proof -
have xm: "x \<in> uminus ` \<Union>{A. \<exists>a. (a, A) \<in> \<D>}"
by (simp add: that)
then obtain a X where "-x \<in> X" "(a, X) \<in> \<D>"
by auto
then show ?thesis
by (metis (no_types, lifting) add.inverse_inverse image_iff pair_imageI)
qed
have 3: "\<And>x X y. \<lbrakk>\<D> tagged_partial_division_of cbox (- b) (- a); (x, X) \<in> \<D>; y \<in> X\<rbrakk> \<Longrightarrow> - y \<in> cbox a b"
by (metis (no_types, lifting) equation_minus_iff imageE subsetD tagged_partial_division_ofD(3) uminus_interval_vector)
have tag': "(\<lambda>(x,K). (- x, uminus ` K)) ` \<D> tagged_division_of cbox a b"
using tag by (auto simp: tagged_division_of_def dest: 1 2 3)
have fine': "\<gamma> fine (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>"
using fine by (fastforce simp: fine_def)
have inj: "inj_on (\<lambda>(x,K). (- x, uminus ` K)) \<D>"
unfolding inj_on_def by force
have eq: "content (uminus ` I) = content I"
if I: "(x, I) \<in> \<D>" and fnz: "f (- x) \<noteq> 0" for x I
proof -
obtain a b where "I = cbox a b"
using tag I that by (force simp: tagged_division_of_def tagged_partial_division_of_def)
then show ?thesis
using content_image_affinity_cbox [of "-1" 0] by auto
qed
have "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>. content K *\<^sub>R f x) =
(\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f (- x))"
apply (simp add: sum.reindex [OF inj])
apply (auto simp: eq intro!: sum.cong)
done
then show ?thesis
using \<gamma> [OF \<open>f \<in> F\<close> tag' fine'] integral_reflect
by (metis (mono_tags, lifting) Henstock_Kurzweil_Integration.integral_cong comp_apply split_def sum.cong)
qed
qed
then show ?thesis
using assms
apply (auto simp: equiintegrable_on_def)
apply (rule integrable_eq)
by auto
qed
subsection%important\<open>Subinterval restrictions for equiintegrable families\<close>
text\<open>First, some technical lemmas about minimizing a "flat" part of a sum over a division.\<close>
lemma lemma0:
assumes "i \<in> Basis"
shows "content (cbox u v) / (interval_upperbound (cbox u v) \<bullet> i - interval_lowerbound (cbox u v) \<bullet> i) =
(if content (cbox u v) = 0 then 0
else \<Prod>j \<in> Basis - {i}. interval_upperbound (cbox u v) \<bullet> j - interval_lowerbound (cbox u v) \<bullet> j)"
proof (cases "content (cbox u v) = 0")
case True
then show ?thesis by simp
next
case False
then show ?thesis
using prod.subset_diff [of "{i}" Basis] assms
by (force simp: content_cbox_if divide_simps split: if_split_asm)
qed
lemma%important content_division_lemma1:
assumes div: "\<D> division_of S" and S: "S \<subseteq> cbox a b" and i: "i \<in> Basis"
and mt: "\<And>K. K \<in> \<D> \<Longrightarrow> content K \<noteq> 0"
and disj: "(\<forall>K \<in> \<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K \<in> \<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})"
shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> content(cbox a b)" (is "?lhs \<le> ?rhs")
proof%unimportant -
have "finite \<D>"
using div by blast
define extend where
"extend \<equiv> \<lambda>K. cbox (\<Sum>j \<in> Basis. if j = i then (a \<bullet> i) *\<^sub>R i else (interval_lowerbound K \<bullet> j) *\<^sub>R j)
(\<Sum>j \<in> Basis. if j = i then (b \<bullet> i) *\<^sub>R i else (interval_upperbound K \<bullet> j) *\<^sub>R j)"
have div_subset_cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b"
using S div by auto
have "\<And>K. K \<in> \<D> \<Longrightarrow> K \<noteq> {}"
using div by blast
have extend: "extend K \<noteq> {}" "extend K \<subseteq> cbox a b" if K: "K \<in> \<D>" for K
proof -
obtain u v where K: "K = cbox u v" "K \<noteq> {}" "K \<subseteq> cbox a b"
using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
with i show "extend K \<noteq> {}" "extend K \<subseteq> cbox a b"
apply (auto simp: extend_def subset_box box_ne_empty sum_if_inner)
by fastforce
qed
have int_extend_disjoint:
"interior(extend K1) \<inter> interior(extend K2) = {}" if K: "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2" for K1 K2
proof -
obtain u v where K1: "K1 = cbox u v" "K1 \<noteq> {}" "K1 \<subseteq> cbox a b"
using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
obtain w z where K2: "K2 = cbox w z" "K2 \<noteq> {}" "K2 \<subseteq> cbox a b"
using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
have cboxes: "cbox u v \<in> \<D>" "cbox w z \<in> \<D>" "cbox u v \<noteq> cbox w z"
using K1 K2 that by auto
with div have "interior (cbox u v) \<inter> interior (cbox w z) = {}"
by blast
moreover
have "\<exists>x. x \<in> box u v \<and> x \<in> box w z"
if "x \<in> interior (extend K1)" "x \<in> interior (extend K2)" for x
proof -
have "a \<bullet> i < x \<bullet> i" "x \<bullet> i < b \<bullet> i"
and ux: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> u \<bullet> k < x \<bullet> k"
and xv: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> x \<bullet> k < v \<bullet> k"
and wx: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> w \<bullet> k < x \<bullet> k"
and xz: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> x \<bullet> k < z \<bullet> k"
using that K1 K2 i by (auto simp: extend_def box_ne_empty sum_if_inner mem_box)
have "box u v \<noteq> {}" "box w z \<noteq> {}"
using cboxes interior_cbox by (auto simp: content_eq_0_interior dest: mt)
then obtain q s
where q: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k < q \<bullet> k \<and> q \<bullet> k < z \<bullet> k"
and s: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k < s \<bullet> k \<and> s \<bullet> k < v \<bullet> k"
by (meson all_not_in_conv mem_box(1))
show ?thesis using disj
proof
assume "\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
then have uva: "(cbox u v) \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
and wza: "(cbox w z) \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
using cboxes by (auto simp: content_eq_0_interior)
then obtain r t where "r \<bullet> i = a \<bullet> i" and r: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k \<le> r \<bullet> k \<and> r \<bullet> k \<le> z \<bullet> k"
and "t \<bullet> i = a \<bullet> i" and t: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k \<le> t \<bullet> k \<and> t \<bullet> k \<le> v \<bullet> k"
by (fastforce simp: mem_box)
have u: "u \<bullet> i < q \<bullet> i"
using i K2(1) K2(3) \<open>t \<bullet> i = a \<bullet> i\<close> q s t [OF i] by (force simp: subset_box)
have w: "w \<bullet> i < s \<bullet> i"
using i K1(1) K1(3) \<open>r \<bullet> i = a \<bullet> i\<close> s r [OF i] by (force simp: subset_box)
let ?x = "(\<Sum>j \<in> Basis. if j = i then min (q \<bullet> i) (s \<bullet> i) *\<^sub>R i else (x \<bullet> j) *\<^sub>R j)"
show ?thesis
proof (intro exI conjI)
show "?x \<in> box u v"
using \<open>i \<in> Basis\<close> s apply (clarsimp simp: mem_box)
apply (subst sum_if_inner; simp)+
apply (fastforce simp: u ux xv)
done
show "?x \<in> box w z"
using \<open>i \<in> Basis\<close> q apply (clarsimp simp: mem_box)
apply (subst sum_if_inner; simp)+
apply (fastforce simp: w wx xz)
done
qed
next
assume "\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
then have uva: "(cbox u v) \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
and wza: "(cbox w z) \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
using cboxes by (auto simp: content_eq_0_interior)
then obtain r t where "r \<bullet> i = b \<bullet> i" and r: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k \<le> r \<bullet> k \<and> r \<bullet> k \<le> z \<bullet> k"
and "t \<bullet> i = b \<bullet> i" and t: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k \<le> t \<bullet> k \<and> t \<bullet> k \<le> v \<bullet> k"
by (fastforce simp: mem_box)
have z: "s \<bullet> i < z \<bullet> i"
using K1(1) K1(3) \<open>r \<bullet> i = b \<bullet> i\<close> r [OF i] i s by (force simp: subset_box)
have v: "q \<bullet> i < v \<bullet> i"
using K2(1) K2(3) \<open>t \<bullet> i = b \<bullet> i\<close> t [OF i] i q by (force simp: subset_box)
let ?x = "(\<Sum>j \<in> Basis. if j = i then max (q \<bullet> i) (s \<bullet> i) *\<^sub>R i else (x \<bullet> j) *\<^sub>R j)"
show ?thesis
proof (intro exI conjI)
show "?x \<in> box u v"
using \<open>i \<in> Basis\<close> s apply (clarsimp simp: mem_box)
apply (subst sum_if_inner; simp)+
apply (fastforce simp: v ux xv)
done
show "?x \<in> box w z"
using \<open>i \<in> Basis\<close> q apply (clarsimp simp: mem_box)
apply (subst sum_if_inner; simp)+
apply (fastforce simp: z wx xz)
done
qed
qed
qed
ultimately show ?thesis by auto
qed
have "?lhs = (\<Sum>K\<in>\<D>. (b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
by (simp add: sum_distrib_left)
also have "\<dots> = sum (content \<circ> extend) \<D>"
proof (rule sum.cong [OF refl])
fix K assume "K \<in> \<D>"
then obtain u v where K: "K = cbox u v" "cbox u v \<noteq> {}" "K \<subseteq> cbox a b"
using cbox_division_memE [OF _ div] div_subset_cbox by metis
then have uv: "u \<bullet> i < v \<bullet> i"
using mt [OF \<open>K \<in> \<D>\<close>] \<open>i \<in> Basis\<close> content_eq_0 by fastforce
have "insert i (Basis \<inter> -{i}) = Basis"
using \<open>i \<in> Basis\<close> by auto
then have "(b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)
= (b \<bullet> i - a \<bullet> i) * (\<Prod>i \<in> insert i (Basis \<inter> -{i}). v \<bullet> i - u \<bullet> i) / (interval_upperbound (cbox u v) \<bullet> i - interval_lowerbound (cbox u v) \<bullet> i)"
using K box_ne_empty(1) content_cbox by fastforce
also have "... = (\<Prod>x\<in>Basis. if x = i then b \<bullet> x - a \<bullet> x
else (interval_upperbound (cbox u v) - interval_lowerbound (cbox u v)) \<bullet> x)"
using \<open>i \<in> Basis\<close> K uv by (simp add: prod.If_cases) (simp add: algebra_simps)
also have "... = (\<Prod>k\<in>Basis.
(\<Sum>j\<in>Basis. if j = i then (b \<bullet> i - a \<bullet> i) *\<^sub>R i else ((interval_upperbound (cbox u v) - interval_lowerbound (cbox u v)) \<bullet> j) *\<^sub>R j) \<bullet> k)"
using \<open>i \<in> Basis\<close> by (subst prod.cong [OF refl sum_if_inner]; simp)
also have "... = (\<Prod>k\<in>Basis.
(\<Sum>j\<in>Basis. if j = i then (b \<bullet> i) *\<^sub>R i else (interval_upperbound (cbox u v) \<bullet> j) *\<^sub>R j) \<bullet> k -
(\<Sum>j\<in>Basis. if j = i then (a \<bullet> i) *\<^sub>R i else (interval_lowerbound (cbox u v) \<bullet> j) *\<^sub>R j) \<bullet> k)"
apply (rule prod.cong [OF refl])
using \<open>i \<in> Basis\<close>
apply (subst sum_if_inner; simp add: algebra_simps)+
done
also have "... = (content \<circ> extend) K"
using \<open>i \<in> Basis\<close> K box_ne_empty
apply (simp add: extend_def)
apply (subst content_cbox, auto)
done
finally show "(b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)
= (content \<circ> extend) K" .
qed
also have "... = sum content (extend ` \<D>)"
proof -
have "\<lbrakk>K1 \<in> \<D>; K2 \<in> \<D>; K1 \<noteq> K2; extend K1 = extend K2\<rbrakk> \<Longrightarrow> content (extend K1) = 0" for K1 K2
using int_extend_disjoint [of K1 K2] extend_def by (simp add: content_eq_0_interior)
then show ?thesis
by (simp add: comm_monoid_add_class.sum.reindex_nontrivial [OF \<open>finite \<D>\<close>])
qed
also have "... \<le> ?rhs"
proof (rule subadditive_content_division)
show "extend ` \<D> division_of UNION \<D> extend"
using int_extend_disjoint apply (auto simp: division_of_def \<open>finite \<D>\<close> extend)
using extend_def apply blast
done
show "UNION \<D> extend \<subseteq> cbox a b"
using extend by fastforce
qed
finally show ?thesis .
qed
proposition%important sum_content_area_over_thin_division:
assumes div: "\<D> division_of S" and S: "S \<subseteq> cbox a b" and i: "i \<in> Basis"
and "a \<bullet> i \<le> c" "c \<le> b \<bullet> i"
and nonmt: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<inter> {x. x \<bullet> i = c} \<noteq> {}"
shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> 2 * content(cbox a b)"
proof%unimportant (cases "content(cbox a b) = 0")
case True
have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) = 0"
using S div by (force intro!: sum.neutral content_0_subset [OF True])
then show ?thesis
by (auto simp: True)
next
case False
then have "content(cbox a b) > 0"
using zero_less_measure_iff by blast
then have "a \<bullet> i < b \<bullet> i" if "i \<in> Basis" for i
using content_pos_lt_eq that by blast
have "finite \<D>"
using div by blast
define Dlec where "Dlec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. x \<bullet> i \<le> c}) ` \<D>. content L \<noteq> 0}"
define Dgec where "Dgec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. x \<bullet> i \<ge> c}) ` \<D>. content L \<noteq> 0}"
define a' where "a' \<equiv> (\<Sum>j\<in>Basis. (if j = i then c else a \<bullet> j) *\<^sub>R j)"
define b' where "b' \<equiv> (\<Sum>j\<in>Basis. (if j = i then c else b \<bullet> j) *\<^sub>R j)"
have Dlec_cbox: "\<And>K. K \<in> Dlec \<Longrightarrow> \<exists>a b. K = cbox a b"
using interval_split [OF i] div by (fastforce simp: Dlec_def division_of_def)
then have lec_is_cbox: "\<lbrakk>content (L \<inter> {x. x \<bullet> i \<le> c}) \<noteq> 0; L \<in> \<D>\<rbrakk> \<Longrightarrow> \<exists>a b. L \<inter> {x. x \<bullet> i \<le> c} = cbox a b" for L
using Dlec_def by blast
have Dgec_cbox: "\<And>K. K \<in> Dgec \<Longrightarrow> \<exists>a b. K = cbox a b"
using interval_split [OF i] div by (fastforce simp: Dgec_def division_of_def)
then have gec_is_cbox: "\<lbrakk>content (L \<inter> {x. x \<bullet> i \<ge> c}) \<noteq> 0; L \<in> \<D>\<rbrakk> \<Longrightarrow> \<exists>a b. L \<inter> {x. x \<bullet> i \<ge> c} = cbox a b" for L
using Dgec_def by blast
have "(b' \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>Dlec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> content(cbox a b')"
proof (rule content_division_lemma1)
show "Dlec division_of \<Union>Dlec"
unfolding division_of_def
proof (intro conjI ballI Dlec_cbox)
show "\<And>K1 K2. \<lbrakk>K1 \<in> Dlec; K2 \<in> Dlec\<rbrakk> \<Longrightarrow> K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}"
by (clarsimp simp: Dlec_def) (use div in auto)
qed (use \<open>finite \<D>\<close> Dlec_def in auto)
show "\<Union>Dlec \<subseteq> cbox a b'"
using Dlec_def div S by (auto simp: b'_def division_of_def mem_box)
show "(\<forall>K\<in>Dlec. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>Dlec. K \<inter> {x. x \<bullet> i = b' \<bullet> i} \<noteq> {})"
using nonmt by (fastforce simp: Dlec_def b'_def sum_if_inner i)
qed (use i Dlec_def in auto)
moreover
have "(\<Sum>K\<in>Dlec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) =
(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)"
apply (subst sum.reindex_nontrivial [OF \<open>finite \<D>\<close>, symmetric], simp)
apply (metis division_split_left_inj [OF div] lec_is_cbox content_eq_0_interior)
unfolding Dlec_def using \<open>finite \<D>\<close> apply (auto simp: sum.mono_neutral_left)
done
moreover have "(b' \<bullet> i - a \<bullet> i) = (c - a \<bullet> i)"
by (simp add: b'_def sum_if_inner i)
ultimately
have lec: "(c - a \<bullet> i) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)
\<le> content(cbox a b')"
by simp
have "(b \<bullet> i - a' \<bullet> i) * (\<Sum>K\<in>Dgec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> content(cbox a' b)"
proof (rule content_division_lemma1)
show "Dgec division_of \<Union>Dgec"
unfolding division_of_def
proof (intro conjI ballI Dgec_cbox)
show "\<And>K1 K2. \<lbrakk>K1 \<in> Dgec; K2 \<in> Dgec\<rbrakk> \<Longrightarrow> K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}"
by (clarsimp simp: Dgec_def) (use div in auto)
qed (use \<open>finite \<D>\<close> Dgec_def in auto)
show "\<Union>Dgec \<subseteq> cbox a' b"
using Dgec_def div S by (auto simp: a'_def division_of_def mem_box)
show "(\<forall>K\<in>Dgec. K \<inter> {x. x \<bullet> i = a' \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>Dgec. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})"
using nonmt by (fastforce simp: Dgec_def a'_def sum_if_inner i)
qed (use i Dgec_def in auto)
moreover
have "(\<Sum>K\<in>Dgec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) =
(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
apply (subst sum.reindex_nontrivial [OF \<open>finite \<D>\<close>, symmetric], simp)
apply (metis division_split_right_inj [OF div] gec_is_cbox content_eq_0_interior)
unfolding Dgec_def using \<open>finite \<D>\<close> apply (auto simp: sum.mono_neutral_left)
done
moreover have "(b \<bullet> i - a' \<bullet> i) = (b \<bullet> i - c)"
by (simp add: a'_def sum_if_inner i)
ultimately
have gec: "(b \<bullet> i - c) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)
\<le> content(cbox a' b)"
by simp
show ?thesis
proof (cases "c = a \<bullet> i \<or> c = b \<bullet> i")
case True
then show ?thesis
proof
assume c: "c = a \<bullet> i"
then have "a' = a"
apply (simp add: sum_if_inner i a'_def cong: if_cong)
using euclidean_representation [of a] sum.cong [OF refl, of Basis "\<lambda>i. (a \<bullet> i) *\<^sub>R i"] by presburger
then have "content (cbox a' b) \<le> 2 * content (cbox a b)" by simp
moreover
have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) /
(interval_upperbound (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) \<bullet> i - interval_lowerbound (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) \<bullet> i))
= (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
(is "sum ?f _ = sum ?g _")
proof (rule sum.cong [OF refl])
fix K assume "K \<in> \<D>"
then have "a \<bullet> i \<le> x \<bullet> i" if "x \<in> K" for x
by (metis S UnionI div division_ofD(6) i mem_box(2) subsetCE that)
then have "K \<inter> {x. a \<bullet> i \<le> x \<bullet> i} = K"
by blast
then show "?f K = ?g K"
by simp
qed
ultimately show ?thesis
using gec c eq by auto
next
assume c: "c = b \<bullet> i"
then have "b' = b"
apply (simp add: sum_if_inner i b'_def cong: if_cong)
using euclidean_representation [of b] sum.cong [OF refl, of Basis "\<lambda>i. (b \<bullet> i) *\<^sub>R i"] by presburger
then have "content (cbox a b') \<le> 2 * content (cbox a b)" by simp
moreover
have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) /
(interval_upperbound (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) \<bullet> i - interval_lowerbound (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) \<bullet> i))
= (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
(is "sum ?f _ = sum ?g _")
proof (rule sum.cong [OF refl])
fix K assume "K \<in> \<D>"
then have "x \<bullet> i \<le> b \<bullet> i" if "x \<in> K" for x
by (metis S UnionI div division_ofD(6) i mem_box(2) subsetCE that)
then have "K \<inter> {x. x \<bullet> i \<le> b \<bullet> i} = K"
by blast
then show "?f K = ?g K"
by simp
qed
ultimately show ?thesis
using lec c eq by auto
qed
next
case False
have prod_if: "(\<Prod>k\<in>Basis \<inter> - {i}. f k) = (\<Prod>k\<in>Basis. f k) / f i" if "f i \<noteq> (0::real)" for f
using that mk_disjoint_insert [OF i]
apply (clarsimp simp add: divide_simps)
by (metis Int_insert_left_if0 finite_Basis finite_insert le_iff_inf mult.commute order_refl prod.insert subset_Compl_singleton)
have abc: "a \<bullet> i < c" "c < b \<bullet> i"
using False assms by auto
then have "(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)
\<le> content(cbox a b') / (c - a \<bullet> i)"
"(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)
\<le> content(cbox a' b) / (b \<bullet> i - c)"
using lec gec by (simp_all add: divide_simps mult.commute)
moreover
have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K) +
(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
(is "?lhs \<le> ?rhs")
proof -
have "?lhs \<le>
(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K +
((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
(is "sum ?f _ \<le> sum ?g _")
proof (rule sum_mono)
fix K assume "K \<in> \<D>"
then obtain u v where uv: "K = cbox u v"
using div by blast
obtain u' v' where uv': "cbox u v \<inter> {x. x \<bullet> i \<le> c} = cbox u v'"
"cbox u v \<inter> {x. c \<le> x \<bullet> i} = cbox u' v"
"\<And>k. k \<in> Basis \<Longrightarrow> u' \<bullet> k = (if k = i then max (u \<bullet> i) c else u \<bullet> k)"
"\<And>k. k \<in> Basis \<Longrightarrow> v' \<bullet> k = (if k = i then min (v \<bullet> i) c else v \<bullet> k)"
using i by (auto simp: interval_split)
have *: "\<lbrakk>content (cbox u v') = 0; content (cbox u' v) = 0\<rbrakk> \<Longrightarrow> content (cbox u v) = 0"
"content (cbox u' v) \<noteq> 0 \<Longrightarrow> content (cbox u v) \<noteq> 0"
"content (cbox u v') \<noteq> 0 \<Longrightarrow> content (cbox u v) \<noteq> 0"
using i uv uv' by (auto simp: content_eq_0 le_max_iff_disj min_le_iff_disj split: if_split_asm intro: order_trans)
show "?f K \<le> ?g K"
using i uv uv' apply (clarsimp simp add: lemma0 * intro!: prod_nonneg)
by (metis content_eq_0 le_less_linear order.strict_implies_order)
qed
also have "... = ?rhs"
by (simp add: sum.distrib)
finally show ?thesis .
qed
moreover have "content (cbox a b') / (c - a \<bullet> i) = content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
using i abc
apply (simp add: field_simps a'_def b'_def measure_lborel_cbox_eq inner_diff)
apply (auto simp: if_distrib if_distrib [of "\<lambda>f. f x" for x] prod.If_cases [of Basis "\<lambda>x. x = i", simplified] prod_if field_simps)
done
moreover have "content (cbox a' b) / (b \<bullet> i - c) = content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
using i abc
apply (simp add: field_simps a'_def b'_def measure_lborel_cbox_eq inner_diff)
apply (auto simp: if_distrib prod.If_cases [of Basis "\<lambda>x. x = i", simplified] prod_if field_simps)
done
ultimately
have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> 2 * content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
by linarith
then show ?thesis
using abc by (simp add: divide_simps mult.commute)
qed
qed
proposition%important bounded_equiintegral_over_thin_tagged_partial_division:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F" and "0 < \<epsilon>"
and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
obtains \<gamma> where "gauge \<gamma>"
"\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b;
\<gamma> fine S; h \<in> F; \<And>x K. (x,K) \<in> S \<Longrightarrow> (K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {})\<rbrakk>
\<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
proof%unimportant (cases "content(cbox a b) = 0")
case True
show ?thesis
proof
show "gauge (\<lambda>x. ball x 1)"
by (simp add: gauge_trivial)
show "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
if "S tagged_partial_division_of cbox a b" "(\<lambda>x. ball x 1) fine S" for S and h:: "'a \<Rightarrow> 'b"
proof -
have "(\<Sum>(x,K) \<in> S. norm (integral K h)) = 0"
using that True content_0_subset
by (fastforce simp: tagged_partial_division_of_def intro: sum.neutral)
with \<open>0 < \<epsilon>\<close> show ?thesis
by simp
qed
qed
next
case False
then have contab_gt0: "content(cbox a b) > 0"
by (simp add: zero_less_measure_iff)
then have a_less_b: "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
by (auto simp: content_pos_lt_eq)
obtain \<gamma>0 where "gauge \<gamma>0"
and \<gamma>0: "\<And>S h. \<lbrakk>S tagged_partial_division_of cbox a b; \<gamma>0 fine S; h \<in> F\<rbrakk>
\<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
proof -
obtain \<gamma> where "gauge \<gamma>"
and \<gamma>: "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
\<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox a b) f)
< \<epsilon>/(5 * (Suc DIM('b)))"
proof -
have e5: "\<epsilon>/(5 * (Suc DIM('b))) > 0"
using \<open>\<epsilon> > 0\<close> by auto
then show ?thesis
using F that by (auto simp: equiintegrable_on_def)
qed
show ?thesis
proof
show "gauge \<gamma>"
by (rule \<open>gauge \<gamma>\<close>)
show "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
if "S tagged_partial_division_of cbox a b" "\<gamma> fine S" "h \<in> F" for S h
proof -
have "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) \<le> 2 * real DIM('b) * (\<epsilon>/(5 * Suc DIM('b)))"
proof (rule Henstock_lemma_part2 [of h a b])
show "h integrable_on cbox a b"
using that F equiintegrable_on_def by metis
show "gauge \<gamma>"
by (rule \<open>gauge \<gamma>\<close>)
qed (use that \<open>\<epsilon> > 0\<close> \<gamma> in auto)
also have "... < \<epsilon>/2"
using \<open>\<epsilon> > 0\<close> by (simp add: divide_simps)
finally show ?thesis .
qed
qed
qed
define \<gamma> where "\<gamma> \<equiv> \<lambda>x. \<gamma>0 x \<inter>
ball x ((\<epsilon>/8 / (norm(f x) + 1)) * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / content(cbox a b))"
have "gauge (\<lambda>x. ball x
(\<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / ((8 * norm (f x) + 8) * content (cbox a b))))"
using \<open>0 < content (cbox a b)\<close> \<open>0 < \<epsilon>\<close> a_less_b
apply (auto simp: gauge_def divide_simps mult_less_0_iff zero_less_mult_iff add_nonneg_eq_0_iff finite_less_Inf_iff)
apply (meson add_nonneg_nonneg mult_nonneg_nonneg norm_ge_zero not_less zero_le_numeral)
done
then have "gauge \<gamma>"
unfolding \<gamma>_def using \<open>gauge \<gamma>0\<close> gauge_Int by auto
moreover
have "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
if "c \<in> cbox a b" "i \<in> Basis" and S: "S tagged_partial_division_of cbox a b"
and "\<gamma> fine S" "h \<in> F" and ne: "\<And>x K. (x,K) \<in> S \<Longrightarrow> K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {}" for c i S h
proof -
have "cbox c b \<subseteq> cbox a b"
by (meson mem_box(2) order_refl subset_box(1) that(1))
have "finite S"
using S by blast
have "\<gamma>0 fine S" and fineS:
"(\<lambda>x. ball x (\<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / ((8 * norm (f x) + 8) * content (cbox a b)))) fine S"
using \<open>\<gamma> fine S\<close> by (auto simp: \<gamma>_def fine_Int)
then have "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
by (intro \<gamma>0 that fineS)
moreover have "(\<Sum>(x,K) \<in> S. norm (integral K h) - norm (content K *\<^sub>R h x - integral K h)) \<le> \<epsilon>/2"
proof -
have "(\<Sum>(x,K) \<in> S. norm (integral K h) - norm (content K *\<^sub>R h x - integral K h))
\<le> (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x))"
proof (clarify intro!: sum_mono)
fix x K
assume xK: "(x,K) \<in> S"
have "norm (integral K h) - norm (content K *\<^sub>R h x - integral K h) \<le> norm (integral K h - (integral K h - content K *\<^sub>R h x))"
by (metis norm_minus_commute norm_triangle_ineq2)
also have "... \<le> norm (content K *\<^sub>R h x)"
by simp
finally show "norm (integral K h) - norm (content K *\<^sub>R h x - integral K h) \<le> norm (content K *\<^sub>R h x)" .
qed
also have "... \<le> (\<Sum>(x,K) \<in> S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
proof (clarify intro!: sum_mono)
fix x K
assume xK: "(x,K) \<in> S"
then have x: "x \<in> cbox a b"
using S unfolding tagged_partial_division_of_def by (meson subset_iff)
let ?\<Delta> = "interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
show "norm (content K *\<^sub>R h x) \<le> \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / ?\<Delta>"
proof (cases "content K = 0")
case True
then show ?thesis by simp
next
case False
then have Kgt0: "content K > 0"
using zero_less_measure_iff by blast
moreover
obtain u v where uv: "K = cbox u v"
using S \<open>(x,K) \<in> S\<close> by blast
then have u_less_v: "\<And>i. i \<in> Basis \<Longrightarrow> u \<bullet> i < v \<bullet> i"
using content_pos_lt_eq uv Kgt0 by blast
then have dist_uv: "dist u v > 0"
using that by auto
ultimately have "norm (h x) \<le> (\<epsilon> * (b \<bullet> i - a \<bullet> i)) / (4 * content (cbox a b) * ?\<Delta>)"
proof -
have "dist x u < \<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
"dist x v < \<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
using fineS u_less_v uv xK
by (force simp: fine_def mem_box field_simps dest!: bspec)+
moreover have "\<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2
\<le> \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
apply (intro mult_left_mono divide_right_mono)
using \<open>i \<in> Basis\<close> \<open>0 < \<epsilon>\<close> apply (auto simp: intro!: cInf_le_finite)
done
ultimately
have "dist x u < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
"dist x v < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
by linarith+
then have duv: "dist u v < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b))"
using dist_triangle_half_r by blast
have uvi: "\<bar>v \<bullet> i - u \<bullet> i\<bar> \<le> norm (v - u)"
by (metis inner_commute inner_diff_right \<open>i \<in> Basis\<close> Basis_le_norm)
have "norm (h x) \<le> norm (f x)"
using x that by (auto simp: norm_f)
also have "... < (norm (f x) + 1)"
by simp
also have "... < \<epsilon> * (b \<bullet> i - a \<bullet> i) / dist u v / (4 * content (cbox a b))"
using duv dist_uv contab_gt0
apply (simp add: divide_simps algebra_simps mult_less_0_iff zero_less_mult_iff split: if_split_asm)
by (meson add_nonneg_nonneg linorder_not_le measure_nonneg mult_nonneg_nonneg norm_ge_zero zero_le_numeral)
also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i) / norm (v - u) / (4 * content (cbox a b))"
by (simp add: dist_norm norm_minus_commute)
also have "... \<le> \<epsilon> * (b \<bullet> i - a \<bullet> i) / \<bar>v \<bullet> i - u \<bullet> i\<bar> / (4 * content (cbox a b))"
apply (intro mult_right_mono divide_left_mono divide_right_mono uvi)
using \<open>0 < \<epsilon>\<close> a_less_b [OF \<open>i \<in> Basis\<close>] u_less_v [OF \<open>i \<in> Basis\<close>] contab_gt0
by (auto simp: less_eq_real_def zero_less_mult_iff that)
also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i)
/ (4 * content (cbox a b) * ?\<Delta>)"
using uv False that(2) u_less_v by fastforce
finally show ?thesis by simp
qed
with Kgt0 have "norm (content K *\<^sub>R h x) \<le> content K * ((\<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b)) / ?\<Delta>)"
using mult_left_mono by fastforce
also have "... = \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
content K / ?\<Delta>"
by (simp add: divide_simps)
finally show ?thesis .
qed
qed
also have "... = (\<Sum>K\<in>snd ` S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K
/ (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
apply (rule sum.over_tagged_division_lemma [OF tagged_partial_division_of_Union_self [OF S]])
apply (simp add: box_eq_empty(1) content_eq_0)
done
also have "... = \<epsilon>/2 * ((b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)))"
by (simp add: sum_distrib_left mult.assoc)
also have "... \<le> (\<epsilon>/2) * 1"
proof (rule mult_left_mono)
have "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
\<le> 2 * content (cbox a b)"
proof (rule sum_content_area_over_thin_division)
show "snd ` S division_of \<Union>(snd ` S)"
by (auto intro: S tagged_partial_division_of_Union_self division_of_tagged_division)
show "UNION S snd \<subseteq> cbox a b"
using S by force
show "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> b \<bullet> i"
using mem_box(2) that by blast+
qed (use that in auto)
then show "(b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> 1"
by (simp add: contab_gt0)
qed (use \<open>0 < \<epsilon>\<close> in auto)
finally show ?thesis by simp
qed
then have "(\<Sum>(x,K) \<in> S. norm (integral K h)) - (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) \<le> \<epsilon>/2"
by (simp add: Groups_Big.sum_subtractf [symmetric])
ultimately show "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
by linarith
qed
ultimately show ?thesis using that by auto
qed
proposition%important equiintegrable_halfspace_restrictions_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F"
and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
shows "(\<Union>i \<in> Basis. \<Union>c. \<Union>h \<in> F. {(\<lambda>x. if x \<bullet> i \<le> c then h x else 0)})
equiintegrable_on cbox a b"
proof%unimportant (cases "content(cbox a b) = 0")
case True
then show ?thesis by simp
next
case False
then have "content(cbox a b) > 0"
using zero_less_measure_iff by blast
then have "a \<bullet> i < b \<bullet> i" if "i \<in> Basis" for i
using content_pos_lt_eq that by blast
have int_F: "f integrable_on cbox a b" if "f \<in> F" for f
using F that by (simp add: equiintegrable_on_def)
let ?CI = "\<lambda>K h x. content K *\<^sub>R h x - integral K h"
show ?thesis
unfolding equiintegrable_on_def
proof (intro conjI; clarify)
show int_lec: "\<lbrakk>i \<in> Basis; h \<in> F\<rbrakk> \<Longrightarrow> (\<lambda>x. if x \<bullet> i \<le> c then h x else 0) integrable_on cbox a b" for i c h
using integrable_restrict_Int [of "{x. x \<bullet> i \<le> c}" h]
apply (auto simp: interval_split Int_commute mem_box intro!: integrable_on_subcbox int_F)
by (metis (full_types, hide_lams) min.bounded_iff)
show "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>f T. f \<in> (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then h x else 0}) \<and>
T tagged_division_of cbox a b \<and> \<gamma> fine T \<longrightarrow>
norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>)"
if "\<epsilon> > 0" for \<epsilon>
proof -
obtain \<gamma>0 where "gauge \<gamma>0" and \<gamma>0:
"\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b;
\<gamma>0 fine S; h \<in> F; \<And>x K. (x,K) \<in> S \<Longrightarrow> (K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {})\<rbrakk>
\<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>/12"
apply (rule bounded_equiintegral_over_thin_tagged_partial_division [OF F f, of \<open>\<epsilon>/12\<close>])
using \<open>\<epsilon> > 0\<close> by (auto simp: norm_f)
obtain \<gamma>1 where "gauge \<gamma>1"
and \<gamma>1: "\<And>h T. \<lbrakk>h \<in> F; T tagged_division_of cbox a b; \<gamma>1 fine T\<rbrakk>
\<Longrightarrow> norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R h x) - integral (cbox a b) h)
< \<epsilon>/(7 * (Suc DIM('b)))"
proof -
have e5: "\<epsilon>/(7 * (Suc DIM('b))) > 0"
using \<open>\<epsilon> > 0\<close> by auto
then show ?thesis
using F that by (auto simp: equiintegrable_on_def)
qed
have h_less3: "(\<Sum>(x,K) \<in> T. norm (?CI K h x)) < \<epsilon>/3"
if "T tagged_partial_division_of cbox a b" "\<gamma>1 fine T" "h \<in> F" for T h
proof -
have "(\<Sum>(x,K) \<in> T. norm (?CI K h x)) \<le> 2 * real DIM('b) * (\<epsilon>/(7 * Suc DIM('b)))"
proof (rule Henstock_lemma_part2 [of h a b])
show "h integrable_on cbox a b"
using that F equiintegrable_on_def by metis
show "gauge \<gamma>1"
by (rule \<open>gauge \<gamma>1\<close>)
qed (use that \<open>\<epsilon> > 0\<close> \<gamma>1 in auto)
also have "... < \<epsilon>/3"
using \<open>\<epsilon> > 0\<close> by (simp add: divide_simps)
finally show ?thesis .
qed
have *: "norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>"
if f: "f = (\<lambda>x. if x \<bullet> i \<le> c then h x else 0)"
and T: "T tagged_division_of cbox a b"
and fine: "(\<lambda>x. \<gamma>0 x \<inter> \<gamma>1 x) fine T" and "i \<in> Basis" "h \<in> F" for f T i c h
proof (cases "a \<bullet> i \<le> c \<and> c \<le> b \<bullet> i")
case True
have "finite T"
using T by blast
define T' where "T' \<equiv> {(x,K) \<in> T. K \<inter> {x. x \<bullet> i \<le> c} \<noteq> {}}"
then have "T' \<subseteq> T"
by auto
then have "finite T'"
using \<open>finite T\<close> infinite_super by blast
have T'_tagged: "T' tagged_partial_division_of cbox a b"
by (meson T \<open>T' \<subseteq> T\<close> tagged_division_of_def tagged_partial_division_subset)
have fine': "\<gamma>0 fine T'" "\<gamma>1 fine T'"
using \<open>T' \<subseteq> T\<close> fine_Int fine_subset fine by blast+
have int_KK': "(\<Sum>(x,K) \<in> T. integral K f) = (\<Sum>(x,K) \<in> T'. integral K f)"
apply (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
using f \<open>finite T\<close> \<open>T' \<subseteq> T\<close>
using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h]
apply (auto simp: T'_def Int_commute)
done
have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = (\<Sum>(x,K) \<in> T'. content K *\<^sub>R f x)"
apply (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
using T f \<open>finite T\<close> \<open>T' \<subseteq> T\<close> apply (force simp: T'_def)
done
moreover have "norm ((\<Sum>(x,K) \<in> T'. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>"
proof -
have *: "norm y < \<epsilon>" if "norm x < \<epsilon>/3" "norm(x - y) \<le> 2 * \<epsilon>/3" for x y::'b
proof -
have "norm y \<le> norm x + norm(x - y)"
by (metis norm_minus_commute norm_triangle_sub)
also have "\<dots> < \<epsilon>/3 + 2*\<epsilon>/3"
using that by linarith
also have "... = \<epsilon>"
by simp
finally show ?thesis .
qed
have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x)
\<le> (\<Sum>(x,K) \<in> T'. norm (?CI K h x))"
by (simp add: norm_sum split_def)
also have "... < \<epsilon>/3"
by (intro h_less3 T'_tagged fine' that)
finally have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x) < \<epsilon>/3" .
moreover have "integral (cbox a b) f = (\<Sum>(x,K) \<in> T. integral K f)"
using int_lec that by (auto simp: integral_combine_tagged_division_topdown)
moreover have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x - ?CI K f x)
\<le> 2*\<epsilon>/3"
proof -
define T'' where "T'' \<equiv> {(x,K) \<in> T'. ~ (K \<subseteq> {x. x \<bullet> i \<le> c})}"
then have "T'' \<subseteq> T'"
by auto
then have "finite T''"
using \<open>finite T'\<close> infinite_super by blast
have T''_tagged: "T'' tagged_partial_division_of cbox a b"
using T'_tagged \<open>T'' \<subseteq> T'\<close> tagged_partial_division_subset by blast
have fine'': "\<gamma>0 fine T''" "\<gamma>1 fine T''"
using \<open>T'' \<subseteq> T'\<close> fine' by (blast intro: fine_subset)+
have "(\<Sum>(x,K) \<in> T'. ?CI K h x - ?CI K f x)
= (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x)"
proof (clarify intro!: sum.mono_neutral_right [OF \<open>finite T'\<close> \<open>T'' \<subseteq> T'\<close>])
fix x K
assume "(x,K) \<in> T'" "(x,K) \<notin> T''"
then have "x \<in> K" "x \<bullet> i \<le> c" "{x. x \<bullet> i \<le> c} \<inter> K = K"
using T''_def T'_tagged by blast+
then show "?CI K h x - ?CI K f x = 0"
using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] by (auto simp: f)
qed
moreover have "norm (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x) \<le> 2*\<epsilon>/3"
proof -
define A where "A \<equiv> {(x,K) \<in> T''. x \<bullet> i \<le> c}"
define B where "B \<equiv> {(x,K) \<in> T''. x \<bullet> i > c}"
then have "A \<subseteq> T''" "B \<subseteq> T''" and disj: "A \<inter> B = {}" and T''_eq: "T'' = A \<union> B"
by (auto simp: A_def B_def)
then have "finite A" "finite B"
using \<open>finite T''\<close> by (auto intro: finite_subset)
have A_tagged: "A tagged_partial_division_of cbox a b"
using T''_tagged \<open>A \<subseteq> T''\<close> tagged_partial_division_subset by blast
have fineA: "\<gamma>0 fine A" "\<gamma>1 fine A"
using \<open>A \<subseteq> T''\<close> fine'' by (blast intro: fine_subset)+
have B_tagged: "B tagged_partial_division_of cbox a b"
using T''_tagged \<open>B \<subseteq> T''\<close> tagged_partial_division_subset by blast
have fineB: "\<gamma>0 fine B" "\<gamma>1 fine B"
using \<open>B \<subseteq> T''\<close> fine'' by (blast intro: fine_subset)+
have "norm (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x)
\<le> (\<Sum>(x,K) \<in> T''. norm (?CI K h x - ?CI K f x))"
by (simp add: norm_sum split_def)
also have "... = (\<Sum>(x,K) \<in> A. norm (?CI K h x - ?CI K f x)) +
(\<Sum>(x,K) \<in> B. norm (?CI K h x - ?CI K f x))"
by (simp add: sum.union_disjoint T''_eq disj \<open>finite A\<close> \<open>finite B\<close>)
also have "... = (\<Sum>(x,K) \<in> A. norm (integral K h - integral K f)) +
(\<Sum>(x,K) \<in> B. norm (?CI K h x + integral K f))"
by (auto simp: A_def B_def f norm_minus_commute intro!: sum.cong arg_cong2 [where f= "(+)"])
also have "... \<le> (\<Sum>(x,K)\<in>A. norm (integral K h)) +
(\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A. norm (integral K h))
+ ((\<Sum>(x,K)\<in>B. norm (?CI K h x)) +
(\<Sum>(x,K)\<in>B. norm (integral K h)) +
(\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h)))"
proof (rule add_mono)
show "(\<Sum>(x,K)\<in>A. norm (integral K h - integral K f))
\<le> (\<Sum>(x,K)\<in>A. norm (integral K h)) +
(\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A.
norm (integral K h))"
proof (subst sum.reindex_nontrivial [OF \<open>finite A\<close>], clarsimp)
fix x K L
assume "(x,K) \<in> A" "(x,L) \<in> A"
and int_ne0: "integral (L \<inter> {x. x \<bullet> i \<le> c}) h \<noteq> 0"
and eq: "K \<inter> {x. x \<bullet> i \<le> c} = L \<inter> {x. x \<bullet> i \<le> c}"
have False if "K \<noteq> L"
proof -
obtain u v where uv: "L = cbox u v"
using T'_tagged \<open>(x, L) \<in> A\<close> \<open>A \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
have "A tagged_division_of UNION A snd"
using A_tagged tagged_partial_division_of_Union_self by auto
then have "interior (K \<inter> {x. x \<bullet> i \<le> c}) = {}"
apply (rule tagged_division_split_left_inj [OF _ \<open>(x,K) \<in> A\<close> \<open>(x,L) \<in> A\<close>])
using that eq \<open>i \<in> Basis\<close> by auto
then show False
using interval_split [OF \<open>i \<in> Basis\<close>] int_ne0 content_eq_0_interior eq uv by fastforce
qed
then show "K = L" by blast
next
show "(\<Sum>(x,K) \<in> A. norm (integral K h - integral K f))
\<le> (\<Sum>(x,K) \<in> A. norm (integral K h)) +
sum ((\<lambda>(x,K). norm (integral K h)) \<circ> (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c}))) A"
using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] f
by (auto simp: Int_commute A_def [symmetric] sum.distrib [symmetric] intro!: sum_mono norm_triangle_ineq4)
qed
next
show "(\<Sum>(x,K)\<in>B. norm (?CI K h x + integral K f))
\<le> (\<Sum>(x,K)\<in>B. norm (?CI K h x)) + (\<Sum>(x,K)\<in>B. norm (integral K h)) +
(\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h))"
proof (subst sum.reindex_nontrivial [OF \<open>finite B\<close>], clarsimp)
fix x K L
assume "(x,K) \<in> B" "(x,L) \<in> B"
and int_ne0: "integral (L \<inter> {x. c \<le> x \<bullet> i}) h \<noteq> 0"
and eq: "K \<inter> {x. c \<le> x \<bullet> i} = L \<inter> {x. c \<le> x \<bullet> i}"
have False if "K \<noteq> L"
proof -
obtain u v where uv: "L = cbox u v"
using T'_tagged \<open>(x, L) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
have "B tagged_division_of UNION B snd"
using B_tagged tagged_partial_division_of_Union_self by auto
then have "interior (K \<inter> {x. c \<le> x \<bullet> i}) = {}"
apply (rule tagged_division_split_right_inj [OF _ \<open>(x,K) \<in> B\<close> \<open>(x,L) \<in> B\<close>])
using that eq \<open>i \<in> Basis\<close> by auto
then show False
using interval_split [OF \<open>i \<in> Basis\<close>] int_ne0
content_eq_0_interior eq uv by fastforce
qed
then show "K = L" by blast
next
show "(\<Sum>(x,K) \<in> B. norm (?CI K h x + integral K f))
\<le> (\<Sum>(x,K) \<in> B. norm (?CI K h x)) +
(\<Sum>(x,K) \<in> B. norm (integral K h)) + sum ((\<lambda>(x,K). norm (integral K h)) \<circ> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i}))) B"
proof (clarsimp simp: B_def [symmetric] sum.distrib [symmetric] intro!: sum_mono)
fix x K
assume "(x,K) \<in> B"
have *: "i = i1 + i2 \<Longrightarrow> norm(c + i1) \<le> norm c + norm i + norm(i2)"
for i::'b and c i1 i2
by (metis add.commute add.left_commute add_diff_cancel_right' dual_order.refl norm_add_rule_thm norm_triangle_ineq4)
obtain u v where uv: "K = cbox u v"
using T'_tagged \<open>(x,K) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
have "h integrable_on cbox a b"
by (simp add: int_F \<open>h \<in> F\<close>)
then have huv: "h integrable_on cbox u v"
apply (rule integrable_on_subcbox)
using B_tagged \<open>(x,K) \<in> B\<close> uv by blast
have "integral K h = integral K f + integral (K \<inter> {x. c \<le> x \<bullet> i}) h"
using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] f uv \<open>i \<in> Basis\<close>
by (simp add: Int_commute integral_split [OF huv \<open>i \<in> Basis\<close>])
then show "norm (?CI K h x + integral K f)
\<le> norm (?CI K h x) + norm (integral K h) + norm (integral (K \<inter> {x. c \<le> x \<bullet> i}) h)"
by (rule *)
qed
qed
qed
also have "... \<le> 2*\<epsilon>/3"
proof -
have overlap: "K \<inter> {x. x \<bullet> i = c} \<noteq> {}" if "(x,K) \<in> T''" for x K
proof -
obtain y y' where y: "y' \<in> K" "c < y' \<bullet> i" "y \<in> K" "y \<bullet> i \<le> c"
using that T''_def T'_def \<open>(x,K) \<in> T''\<close> by fastforce
obtain u v where uv: "K = cbox u v"
using T''_tagged \<open>(x,K) \<in> T''\<close> by blast
then have "connected K"
by (simp add: is_interval_cbox is_interval_connected)
then have "(\<exists>z \<in> K. z \<bullet> i = c)"
using y connected_ivt_component by fastforce
then show ?thesis
by fastforce
qed
have **: "\<lbrakk>x < \<epsilon>/12; y < \<epsilon>/12; z \<le> \<epsilon>/2\<rbrakk> \<Longrightarrow> x + y + z \<le> 2 * \<epsilon>/3" for x y z
by auto
show ?thesis
proof (rule **)
have cb_ab: "(\<Sum>j \<in> Basis. if j = i then c *\<^sub>R i else (a \<bullet> j) *\<^sub>R j) \<in> cbox a b"
using \<open>i \<in> Basis\<close> True \<open>\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i < b \<bullet> i\<close>
apply (clarsimp simp add: mem_box)
apply (subst sum_if_inner | force)+
done
show "(\<Sum>(x,K) \<in> A. norm (integral K h)) < \<epsilon>/12"
apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> A_tagged fineA(1) \<open>h \<in> F\<close>])
using \<open>i \<in> Basis\<close> \<open>A \<subseteq> T''\<close> overlap
apply (subst sum_if_inner | force)+
done
have 1: "(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A tagged_partial_division_of cbox a b"
using \<open>finite A\<close> \<open>i \<in> Basis\<close>
apply (auto simp: tagged_partial_division_of_def)
using A_tagged apply (auto simp: A_def)
using interval_split(1) by blast
have 2: "\<gamma>0 fine (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A"
using fineA(1) fine_def by fastforce
show "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A. norm (integral K h)) < \<epsilon>/12"
apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
using \<open>i \<in> Basis\<close> apply (subst sum_if_inner | force)+
using overlap apply (auto simp: A_def)
done
have *: "\<lbrakk>x < \<epsilon>/3; y < \<epsilon>/12; z < \<epsilon>/12\<rbrakk> \<Longrightarrow> x + y + z \<le> \<epsilon>/2" for x y z
by auto
show "(\<Sum>(x,K) \<in> B. norm (?CI K h x)) +
(\<Sum>(x,K) \<in> B. norm (integral K h)) +
(\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h))
\<le> \<epsilon>/2"
proof (rule *)
show "(\<Sum>(x,K) \<in> B. norm (?CI K h x)) < \<epsilon>/3"
by (intro h_less3 B_tagged fineB that)
show "(\<Sum>(x,K) \<in> B. norm (integral K h)) < \<epsilon>/12"
apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> B_tagged fineB(1) \<open>h \<in> F\<close>])
using \<open>i \<in> Basis\<close> \<open>B \<subseteq> T''\<close> overlap by (subst sum_if_inner | force)+
have 1: "(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B tagged_partial_division_of cbox a b"
using \<open>finite B\<close> \<open>i \<in> Basis\<close>
apply (auto simp: tagged_partial_division_of_def)
using B_tagged apply (auto simp: B_def)
using interval_split(2) by blast
have 2: "\<gamma>0 fine (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B"
using fineB(1) fine_def by fastforce
show "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h)) < \<epsilon>/12"
apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
using \<open>i \<in> Basis\<close> apply (subst sum_if_inner | force)+
using overlap apply (auto simp: B_def)
done
qed
qed
qed
finally show ?thesis .
qed
ultimately show ?thesis by metis
qed
ultimately show ?thesis
by (simp add: sum_subtractf [symmetric] int_KK' *)
qed
ultimately show ?thesis by metis
next
case False
then consider "c < a \<bullet> i" | "b \<bullet> i < c"
by auto
then show ?thesis
proof cases
case 1
then have f0: "f x = 0" if "x \<in> cbox a b" for x
using that f \<open>i \<in> Basis\<close> mem_box(2) by force
then have int_f0: "integral (cbox a b) f = 0"
by (simp add: integral_cong)
have f0_tag: "f x = 0" if "(x,K) \<in> T" for x K
using T f0 that by (force simp: tagged_division_of_def)
then have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = 0"
by (metis (mono_tags, lifting) real_vector.scale_eq_0_iff split_conv sum.neutral surj_pair)
then show ?thesis
using \<open>0 < \<epsilon>\<close> by (simp add: int_f0)
next
case 2
then have fh: "f x = h x" if "x \<in> cbox a b" for x
using that f \<open>i \<in> Basis\<close> mem_box(2) by force
then have int_f: "integral (cbox a b) f = integral (cbox a b) h"
using integral_cong by blast
have fh_tag: "f x = h x" if "(x,K) \<in> T" for x K
using T fh that by (force simp: tagged_division_of_def)
then have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = (\<Sum>(x,K) \<in> T. content K *\<^sub>R h x)"
by (metis (mono_tags, lifting) split_cong sum.cong)
with \<open>0 < \<epsilon>\<close> show ?thesis
apply (simp add: int_f)
apply (rule less_trans [OF \<gamma>1])
using that fine_Int apply (force simp: divide_simps)+
done
qed
qed
have "gauge (\<lambda>x. \<gamma>0 x \<inter> \<gamma>1 x)"
by (simp add: \<open>gauge \<gamma>0\<close> \<open>gauge \<gamma>1\<close> gauge_Int)
then show ?thesis
by (auto intro: *)
qed
qed
qed
corollary%important equiintegrable_halfspace_restrictions_ge:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F"
and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
shows "(\<Union>i \<in> Basis. \<Union>c. \<Union>h \<in> F. {(\<lambda>x. if x \<bullet> i \<ge> c then h x else 0)})
equiintegrable_on cbox a b"
proof%unimportant -
have *: "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>(\<lambda>f. f \<circ> uminus) ` F. {\<lambda>x. if x \<bullet> i \<le> c then h x else 0})
equiintegrable_on cbox (- b) (- a)"
proof (rule equiintegrable_halfspace_restrictions_le)
show "(\<lambda>f. f \<circ> uminus) ` F equiintegrable_on cbox (- b) (- a)"
using F equiintegrable_reflect by blast
show "f \<circ> uminus \<in> (\<lambda>f. f \<circ> uminus) ` F"
using f by auto
show "\<And>h x. \<lbrakk>h \<in> (\<lambda>f. f \<circ> uminus) ` F; x \<in> cbox (- b) (- a)\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm ((f \<circ> uminus) x)"
using f apply (clarsimp simp:)
by (metis add.inverse_inverse image_eqI norm_f uminus_interval_vector)
qed
have eq: "(\<lambda>f. f \<circ> uminus) `
(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then (h \<circ> uminus) x else 0}) =
(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else 0})"
apply (auto simp: o_def cong: if_cong)
using minus_le_iff apply fastforce
apply (rule_tac x="\<lambda>x. if c \<le> (-x) \<bullet> i then h(-x) else 0" in image_eqI)
using le_minus_iff apply fastforce+
done
show ?thesis
using equiintegrable_reflect [OF *] by (auto simp: eq)
qed
proposition%important equiintegrable_closed_interval_restrictions:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes f: "f integrable_on cbox a b"
shows "(\<Union>c d. {(\<lambda>x. if x \<in> cbox c d then f x else 0)}) equiintegrable_on cbox a b"
proof%unimportant -
let ?g = "\<lambda>B c d x. if \<forall>i\<in>B. c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i then f x else 0"
have *: "insert f (\<Union>c d. {?g B c d}) equiintegrable_on cbox a b" if "B \<subseteq> Basis" for B
proof -
have "finite B"
using finite_Basis finite_subset \<open>B \<subseteq> Basis\<close> by blast
then show ?thesis using \<open>B \<subseteq> Basis\<close>
proof (induction B)
case empty
with f show ?case by auto
next
case (insert i B)
then have "i \<in> Basis"
by auto
have *: "norm (h x) \<le> norm (f x)"
if "h \<in> insert f (\<Union>c d. {?g B c d})" "x \<in> cbox a b" for h x
using that by auto
have "(\<Union>i\<in>Basis.
\<Union>\<xi>. \<Union>h\<in>insert f (\<Union>i\<in>Basis. \<Union>\<psi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<psi> then h x else 0}).
{\<lambda>x. if \<xi> \<le> x \<bullet> i then h x else 0})
equiintegrable_on cbox a b"
proof (rule equiintegrable_halfspace_restrictions_ge [where f=f])
show "insert f (\<Union>i\<in>Basis. \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}).
{\<lambda>x. if x \<bullet> i \<le> \<xi> then h x else 0}) equiintegrable_on cbox a b"
apply (intro * f equiintegrable_on_insert equiintegrable_halfspace_restrictions_le [OF insert.IH insertI1])
using insert.prems apply auto
done
show"norm(h x) \<le> norm(f x)"
if "h \<in> insert f (\<Union>i\<in>Basis. \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<xi> then h x else 0})"
"x \<in> cbox a b" for h x
using that by auto
qed auto
then have "insert f (\<Union>i\<in>Basis.
\<Union>\<xi>. \<Union>h\<in>insert f (\<Union>i\<in>Basis. \<Union>\<psi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<psi> then h x else 0}).
{\<lambda>x. if \<xi> \<le> x \<bullet> i then h x else 0})
equiintegrable_on cbox a b"
by (blast intro: f equiintegrable_on_insert)
then show ?case
apply (rule equiintegrable_on_subset, clarify)
using \<open>i \<in> Basis\<close> apply simp
apply (drule_tac x=i in bspec, assumption)
apply (drule_tac x="c \<bullet> i" in spec, clarify)
apply (drule_tac x=i in bspec, assumption)
apply (drule_tac x="d \<bullet> i" in spec)
apply (clarsimp simp add: fun_eq_iff)
apply (drule_tac x=c in spec)
apply (drule_tac x=d in spec)
apply (simp add: split: if_split_asm)
done
qed
qed
show ?thesis
by (rule equiintegrable_on_subset [OF * [OF subset_refl]]) (auto simp: mem_box)
qed
subsection%important\<open>Continuity of the indefinite integral\<close>
proposition%important indefinite_integral_continuous:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes int_f: "f integrable_on cbox a b"
and c: "c \<in> cbox a b" and d: "d \<in> cbox a b" "0 < \<epsilon>"
obtains \<delta> where "0 < \<delta>"
"\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm(c' - c) \<le> \<delta>; norm(d' - d) \<le> \<delta>\<rbrakk>
\<Longrightarrow> norm(integral(cbox c' d') f - integral(cbox c d) f) < \<epsilon>"
proof%unimportant -
{ assume "\<exists>c' d'. c' \<in> cbox a b \<and> d' \<in> cbox a b \<and> norm(c' - c) \<le> \<delta> \<and> norm(d' - d) \<le> \<delta> \<and>
norm(integral(cbox c' d') f - integral(cbox c d) f) \<ge> \<epsilon>"
(is "\<exists>c' d'. ?\<Phi> c' d' \<delta>") if "0 < \<delta>" for \<delta>
then have "\<exists>c' d'. ?\<Phi> c' d' (1 / Suc n)" for n
by simp
then obtain u v where "\<And>n. ?\<Phi> (u n) (v n) (1 / Suc n)"
by metis
then have u: "u n \<in> cbox a b" and norm_u: "norm(u n - c) \<le> 1 / Suc n"
and v: "v n \<in> cbox a b" and norm_v: "norm(v n - d) \<le> 1 / Suc n"
and \<epsilon>: "\<epsilon> \<le> norm (integral (cbox (u n) (v n)) f - integral (cbox c d) f)" for n
by blast+
then have False
proof -
have uvn: "cbox (u n) (v n) \<subseteq> cbox a b" for n
by (meson u v mem_box(2) subset_box(1))
define S where "S \<equiv> \<Union>i \<in> Basis. {x. x \<bullet> i = c \<bullet> i} \<union> {x. x \<bullet> i = d \<bullet> i}"
have "negligible S"
unfolding S_def by force
then have int_f': "(\<lambda>x. if x \<in> S then 0 else f x) integrable_on cbox a b"
by (force intro: integrable_spike assms)
have get_n: "\<exists>n. \<forall>m\<ge>n. x \<in> cbox (u m) (v m) \<longleftrightarrow> x \<in> cbox c d" if x: "x \<notin> S" for x
proof -
define \<epsilon> where "\<epsilon> \<equiv> Min ((\<lambda>i. min \<bar>x \<bullet> i - c \<bullet> i\<bar> \<bar>x \<bullet> i - d \<bullet> i\<bar>) ` Basis)"
have "\<epsilon> > 0"
using \<open>x \<notin> S\<close> by (auto simp: S_def \<epsilon>_def)
then obtain n where "n \<noteq> 0" and n: "1 / (real n) < \<epsilon>"
by (metis inverse_eq_divide real_arch_inverse)
have emin: "\<epsilon> \<le> min \<bar>x \<bullet> i - c \<bullet> i\<bar> \<bar>x \<bullet> i - d \<bullet> i\<bar>" if "i \<in> Basis" for i
unfolding \<epsilon>_def
apply (rule Min.coboundedI)
using that by force+
have "1 / real (Suc n) < \<epsilon>"
using n \<open>n \<noteq> 0\<close> \<open>\<epsilon> > 0\<close> by (simp add: field_simps)
have "x \<in> cbox (u m) (v m) \<longleftrightarrow> x \<in> cbox c d" if "m \<ge> n" for m
proof -
have *: "\<lbrakk>\<bar>u - c\<bar> \<le> n; \<bar>v - d\<bar> \<le> n; N < \<bar>x - c\<bar>; N < \<bar>x - d\<bar>; n \<le> N\<rbrakk>
\<Longrightarrow> u \<le> x \<and> x \<le> v \<longleftrightarrow> c \<le> x \<and> x \<le> d" for N n u v c d and x::real
by linarith
have "(u m \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> v m \<bullet> i) = (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i)"
if "i \<in> Basis" for i
proof (rule *)
show "\<bar>u m \<bullet> i - c \<bullet> i\<bar> \<le> 1 / Suc m"
using norm_u [of m]
by (metis (full_types) order_trans Basis_le_norm inner_commute inner_diff_right that)
show "\<bar>v m \<bullet> i - d \<bullet> i\<bar> \<le> 1 / real (Suc m)"
using norm_v [of m]
by (metis (full_types) order_trans Basis_le_norm inner_commute inner_diff_right that)
show "1/n < \<bar>x \<bullet> i - c \<bullet> i\<bar>" "1/n < \<bar>x \<bullet> i - d \<bullet> i\<bar>"
using n \<open>n \<noteq> 0\<close> emin [OF \<open>i \<in> Basis\<close>]
by (simp_all add: inverse_eq_divide)
show "1 / real (Suc m) \<le> 1 / real n"
using \<open>n \<noteq> 0\<close> \<open>m \<ge> n\<close> by (simp add: divide_simps)
qed
then show ?thesis by (simp add: mem_box)
qed
then show ?thesis by blast
qed
have 1: "range (\<lambda>n x. if x \<in> cbox (u n) (v n) then if x \<in> S then 0 else f x else 0) equiintegrable_on cbox a b"
by (blast intro: equiintegrable_on_subset [OF equiintegrable_closed_interval_restrictions [OF int_f']])
have 2: "(\<lambda>n. if x \<in> cbox (u n) (v n) then if x \<in> S then 0 else f x else 0)
\<longlonglongrightarrow> (if x \<in> cbox c d then if x \<in> S then 0 else f x else 0)" for x
by (fastforce simp: dest: get_n intro: Lim_eventually eventually_sequentiallyI)
have [simp]: "cbox c d \<inter> cbox a b = cbox c d"
using c d by (force simp: mem_box)
have [simp]: "cbox (u n) (v n) \<inter> cbox a b = cbox (u n) (v n)" for n
using u v by (fastforce simp: mem_box intro: order.trans)
have "\<And>y A. y \<in> A - S \<Longrightarrow> f y = (\<lambda>x. if x \<in> S then 0 else f x) y"
by simp
then have "\<And>A. integral A (\<lambda>x. if x \<in> S then 0 else f (x)) = integral A (\<lambda>x. f (x))"
by (blast intro: integral_spike [OF \<open>negligible S\<close>])
moreover
obtain N where "dist (integral (cbox (u N) (v N)) (\<lambda>x. if x \<in> S then 0 else f x))
(integral (cbox c d) (\<lambda>x. if x \<in> S then 0 else f x)) < \<epsilon>"
using equiintegrable_limit [OF 1 2] \<open>0 < \<epsilon>\<close> by (force simp: integral_restrict_Int lim_sequentially)
ultimately have "dist (integral (cbox (u N) (v N)) f) (integral (cbox c d) f) < \<epsilon>"
by simp
then show False
by (metis dist_norm not_le \<epsilon>)
qed
}
then show ?thesis
by (meson not_le that)
qed
corollary%important indefinite_integral_uniformly_continuous:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "f integrable_on cbox a b"
shows "uniformly_continuous_on (cbox (Pair a a) (Pair b b)) (\<lambda>y. integral (cbox (fst y) (snd y)) f)"
proof%unimportant -
show ?thesis
proof (rule compact_uniformly_continuous, clarsimp simp add: continuous_on_iff)
fix c d and \<epsilon>::real
assume c: "c \<in> cbox a b" and d: "d \<in> cbox a b" and "0 < \<epsilon>"
obtain \<delta> where "0 < \<delta>" and \<delta>:
"\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm(c' - c) \<le> \<delta>; norm(d' - d) \<le> \<delta>\<rbrakk>
\<Longrightarrow> norm(integral(cbox c' d') f -
integral(cbox c d) f) < \<epsilon>"
using indefinite_integral_continuous \<open>0 < \<epsilon>\<close> assms c d by blast
show "\<exists>\<delta> > 0. \<forall>x' \<in> cbox (a, a) (b, b).
dist x' (c, d) < \<delta> \<longrightarrow>
dist (integral (cbox (fst x') (snd x')) f)
(integral (cbox c d) f)
< \<epsilon>"
using \<open>0 < \<delta>\<close>
by (force simp: dist_norm intro: \<delta> order_trans [OF norm_fst_le] order_trans [OF norm_snd_le] less_imp_le)
qed auto
qed
corollary%important bounded_integrals_over_subintervals:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
assumes "f integrable_on cbox a b"
shows "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> cbox a b}"
proof%unimportant -
have "bounded ((\<lambda>y. integral (cbox (fst y) (snd y)) f) ` cbox (a, a) (b, b))"
(is "bounded ?I")
by (blast intro: bounded_cbox bounded_uniformly_continuous_image indefinite_integral_uniformly_continuous [OF assms])
then obtain B where "B > 0" and B: "\<And>x. x \<in> ?I \<Longrightarrow> norm x \<le> B"
by (auto simp: bounded_pos)
have "norm x \<le> B" if "x = integral (cbox c d) f" "cbox c d \<subseteq> cbox a b" for x c d
proof (cases "cbox c d = {}")
case True
with \<open>0 < B\<close> that show ?thesis by auto
next
case False
show ?thesis
apply (rule B)
using that \<open>B > 0\<close> False apply (clarsimp simp: image_def)
by (metis cbox_Pair_iff interval_subset_is_interval is_interval_cbox prod.sel)
qed
then show ?thesis
by (blast intro: boundedI)
qed
text\<open>An existence theorem for "improper" integrals.
Hake's theorem implies that if the integrals over subintervals have a limit, the integral exists.
We only need to assume that the integrals are bounded, and we get absolute integrability,
but we also need a (rather weak) bound assumption on the function.\<close>
theorem%important absolutely_integrable_improper:
fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
assumes int_f: "\<And>c d. cbox c d \<subseteq> box a b \<Longrightarrow> f integrable_on cbox c d"
and bo: "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> box a b}"
and absi: "\<And>i. i \<in> Basis
\<Longrightarrow> \<exists>g. g absolutely_integrable_on cbox a b \<and>
((\<forall>x \<in> cbox a b. f x \<bullet> i \<le> g x) \<or> (\<forall>x \<in> cbox a b. f x \<bullet> i \<ge> g x))"
shows "f absolutely_integrable_on cbox a b"
proof%unimportant (cases "content(cbox a b) = 0")
case True
then show ?thesis
by auto
next
case False
then have pos: "content(cbox a b) > 0"
using zero_less_measure_iff by blast
show ?thesis
unfolding absolutely_integrable_componentwise_iff [where f = f]
proof
fix j::'N
assume "j \<in> Basis"
then obtain g where absint_g: "g absolutely_integrable_on cbox a b"
and g: "(\<forall>x \<in> cbox a b. f x \<bullet> j \<le> g x) \<or> (\<forall>x \<in> cbox a b. f x \<bullet> j \<ge> g x)"
using absi by blast
have int_gab: "g integrable_on cbox a b"
using absint_g set_lebesgue_integral_eq_integral(1) by blast
have 1: "cbox (a + (b - a) /\<^sub>R real (Suc n)) (b - (b - a) /\<^sub>R real (Suc n)) \<subseteq> box a b" for n
apply (rule subset_box_imp)
using pos apply (auto simp: content_pos_lt_eq algebra_simps)
done
have 2: "cbox (a + (b - a) /\<^sub>R real (Suc n)) (b - (b - a) /\<^sub>R real (Suc n)) \<subseteq>
cbox (a + (b - a) /\<^sub>R real (Suc n + 1)) (b - (b - a) /\<^sub>R real (Suc n + 1))" for n
apply (rule subset_box_imp)
using pos apply (simp add: content_pos_lt_eq algebra_simps)
apply (simp add: divide_simps)
apply (auto simp: field_simps)
done
have getN: "\<exists>N::nat. \<forall>k. k \<ge> N \<longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
if x: "x \<in> box a b" for x
proof -
let ?\<Delta> = "(\<Union>i \<in> Basis. {((x - a) \<bullet> i) / ((b - a) \<bullet> i), (b - x) \<bullet> i / ((b - a) \<bullet> i)})"
obtain N where N: "real N > 1 / Inf ?\<Delta>"
using reals_Archimedean2 by blast
moreover have \<Delta>: "Inf ?\<Delta> > 0"
using that by (auto simp: finite_less_Inf_iff mem_box algebra_simps divide_simps)
ultimately have "N > 0"
using of_nat_0_less_iff by fastforce
show ?thesis
proof (intro exI impI allI)
fix k assume "N \<le> k"
with \<open>0 < N\<close> have "k > 0"
by linarith
have xa_gt: "(x - a) \<bullet> i > ((b - a) \<bullet> i) / (real k)" if "i \<in> Basis" for i
proof -
have *: "Inf ?\<Delta> \<le> ((x - a) \<bullet> i) / ((b - a) \<bullet> i)"
using that by (force intro: cInf_le_finite)
have "1 / Inf ?\<Delta> \<ge> ((b - a) \<bullet> i) / ((x - a) \<bullet> i)"
using le_imp_inverse_le [OF * \<Delta>]
by (simp add: field_simps)
with N have "k > ((b - a) \<bullet> i) / ((x - a) \<bullet> i)"
using \<open>N \<le> k\<close> by linarith
with x that show ?thesis
by (auto simp: mem_box algebra_simps divide_simps)
qed
have bx_gt: "(b - x) \<bullet> i > ((b - a) \<bullet> i) / k" if "i \<in> Basis" for i
proof -
have *: "Inf ?\<Delta> \<le> ((b - x) \<bullet> i) / ((b - a) \<bullet> i)"
using that by (force intro: cInf_le_finite)
have "1 / Inf ?\<Delta> \<ge> ((b - a) \<bullet> i) / ((b - x) \<bullet> i)"
using le_imp_inverse_le [OF * \<Delta>]
by (simp add: field_simps)
with N have "k > ((b - a) \<bullet> i) / ((b - x) \<bullet> i)"
using \<open>N \<le> k\<close> by linarith
with x that show ?thesis
by (auto simp: mem_box algebra_simps divide_simps)
qed
show "x \<in> cbox (a + (b - a) /\<^sub>R k) (b - (b - a) /\<^sub>R k)"
using that \<Delta> \<open>k > 0\<close>
by (auto simp: mem_box algebra_simps divide_inverse dest: xa_gt bx_gt)
qed
qed
obtain Bf where "Bf > 0" and Bf: "\<And>c d. cbox c d \<subseteq> box a b \<Longrightarrow> norm (integral (cbox c d) f) \<le> Bf"
using bo unfolding bounded_pos by blast
obtain Bg where "Bg > 0" and Bg:"\<And>c d. cbox c d \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox c d) g\<bar> \<le> Bg"
using bounded_integrals_over_subintervals [OF int_gab] unfolding bounded_pos real_norm_def by blast
show "(\<lambda>x. f x \<bullet> j) absolutely_integrable_on cbox a b"
using g
proof \<comment> \<open>A lot of duplication in the two proofs\<close>
assume fg [rule_format]: "\<forall>x\<in>cbox a b. f x \<bullet> j \<le> g x"
have "(\<lambda>x. (f x \<bullet> j)) = (\<lambda>x. g x - (g x - (f x \<bullet> j)))"
by simp
moreover have "(\<lambda>x. g x - (g x - (f x \<bullet> j))) integrable_on cbox a b"
proof (rule Henstock_Kurzweil_Integration.integrable_diff [OF int_gab])
let ?\<phi> = "\<lambda>k x. if x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))
then g x - f x \<bullet> j else 0"
have "(\<lambda>x. g x - f x \<bullet> j) integrable_on box a b"
proof (rule monotone_convergence_increasing [of ?\<phi>, THEN conjunct1])
have *: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k)) \<inter> box a b
= cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))" for k
using box_subset_cbox "1" by fastforce
show "?\<phi> k integrable_on box a b" for k
apply (simp add: integrable_restrict_Int integral_restrict_Int *)
apply (rule integrable_diff [OF integrable_on_subcbox [OF int_gab]])
using "*" box_subset_cbox apply blast
by (metis "1" int_f integrable_component of_nat_Suc)
have cb12: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
\<subseteq> cbox (a + (b - a) /\<^sub>R (2 + real k)) (b - (b - a) /\<^sub>R (2 + real k))" for k
using False content_eq_0
apply (simp add: subset_box algebra_simps)
apply (simp add: divide_simps)
apply (fastforce simp: field_simps)
done
show "?\<phi> k x \<le> ?\<phi> (Suc k) x" if "x \<in> box a b" for k x
using cb12 box_subset_cbox that by (force simp: intro!: fg)
show "(\<lambda>k. ?\<phi> k x) \<longlonglongrightarrow> g x - f x \<bullet> j" if x: "x \<in> box a b" for x
proof (rule Lim_eventually)
obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
using getN [OF x] by blast
show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = g x - f x \<bullet> j"
proof
fix k::nat assume "N \<le> k"
have "x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))"
by (metis \<open>N \<le> k\<close> le_Suc_eq N)
then show "?\<phi> k x = g x - f x \<bullet> j"
by simp
qed
qed
have "\<bar>integral (box a b)
(\<lambda>x. if x \<in> cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
then g x - f x \<bullet> j else 0)\<bar> \<le> Bg + Bf" for k
proof -
let ?I = "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))"
have I_int [simp]: "?I \<inter> box a b = ?I"
using 1 by (simp add: Int_absorb2)
have int_fI: "f integrable_on ?I"
apply (rule integrable_subinterval [OF int_f order_refl])
using "*" box_subset_cbox by blast
then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
by (simp add: integrable_component)
moreover have "g integrable_on ?I"
apply (rule integrable_subinterval [OF int_gab])
using "*" box_subset_cbox by blast
moreover
have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
with 1 I_int have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
by (blast intro: order_trans [OF _ Bf])
ultimately show ?thesis
apply (simp add: integral_restrict_Int integral_diff)
using "*" box_subset_cbox by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
qed
then show "bounded (range (\<lambda>k. integral (box a b) (?\<phi> k)))"
apply (simp add: bounded_pos)
apply (rule_tac x="Bg+Bf" in exI)
using \<open>0 < Bf\<close> \<open>0 < Bg\<close> apply auto
done
qed
then show "(\<lambda>x. g x - f x \<bullet> j) integrable_on cbox a b"
by (simp add: integrable_on_open_interval)
qed
ultimately have "(\<lambda>x. f x \<bullet> j) integrable_on cbox a b"
by auto
then show ?thesis
apply (rule absolutely_integrable_component_ubound [OF _ absint_g])
by (simp add: fg)
next
assume gf [rule_format]: "\<forall>x\<in>cbox a b. g x \<le> f x \<bullet> j"
have "(\<lambda>x. (f x \<bullet> j)) = (\<lambda>x. ((f x \<bullet> j) - g x) + g x)"
by simp
moreover have "(\<lambda>x. (f x \<bullet> j - g x) + g x) integrable_on cbox a b"
proof (rule Henstock_Kurzweil_Integration.integrable_add [OF _ int_gab])
let ?\<phi> = "\<lambda>k x. if x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))
then f x \<bullet> j - g x else 0"
have "(\<lambda>x. f x \<bullet> j - g x) integrable_on box a b"
proof (rule monotone_convergence_increasing [of ?\<phi>, THEN conjunct1])
have *: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k)) \<inter> box a b
= cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))" for k
using box_subset_cbox "1" by fastforce
show "?\<phi> k integrable_on box a b" for k
apply (simp add: integrable_restrict_Int integral_restrict_Int *)
apply (rule integrable_diff)
apply (metis "1" int_f integrable_component of_nat_Suc)
apply (rule integrable_on_subcbox [OF int_gab])
using "*" box_subset_cbox apply blast
done
have cb12: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
\<subseteq> cbox (a + (b - a) /\<^sub>R (2 + real k)) (b - (b - a) /\<^sub>R (2 + real k))" for k
using False content_eq_0
apply (simp add: subset_box algebra_simps)
apply (simp add: divide_simps)
apply (fastforce simp: field_simps)
done
show "?\<phi> k x \<le> ?\<phi> (Suc k) x" if "x \<in> box a b" for k x
using cb12 box_subset_cbox that by (force simp: intro!: gf)
show "(\<lambda>k. ?\<phi> k x) \<longlonglongrightarrow> f x \<bullet> j - g x" if x: "x \<in> box a b" for x
proof (rule Lim_eventually)
obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
using getN [OF x] by blast
show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = f x \<bullet> j - g x"
proof
fix k::nat assume "N \<le> k"
have "x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))"
by (metis \<open>N \<le> k\<close> le_Suc_eq N)
then show "?\<phi> k x = f x \<bullet> j - g x"
by simp
qed
qed
have "\<bar>integral (box a b)
(\<lambda>x. if x \<in> cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
then f x \<bullet> j - g x else 0)\<bar> \<le> Bf + Bg" for k
proof -
let ?I = "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))"
have I_int [simp]: "?I \<inter> box a b = ?I"
using 1 by (simp add: Int_absorb2)
have int_fI: "f integrable_on ?I"
apply (rule integrable_subinterval [OF int_f order_refl])
using "*" box_subset_cbox by blast
then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
by (simp add: integrable_component)
moreover have "g integrable_on ?I"
apply (rule integrable_subinterval [OF int_gab])
using "*" box_subset_cbox by blast
moreover
have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
with 1 I_int have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
by (blast intro: order_trans [OF _ Bf])
ultimately show ?thesis
apply (simp add: integral_restrict_Int integral_diff)
using "*" box_subset_cbox by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
qed
then show "bounded (range (\<lambda>k. integral (box a b) (?\<phi> k)))"
apply (simp add: bounded_pos)
apply (rule_tac x="Bf+Bg" in exI)
using \<open>0 < Bf\<close> \<open>0 < Bg\<close> by auto
qed
then show "(\<lambda>x. f x \<bullet> j - g x) integrable_on cbox a b"
by (simp add: integrable_on_open_interval)
qed
ultimately have "(\<lambda>x. f x \<bullet> j) integrable_on cbox a b"
by auto
then show ?thesis
apply (rule absolutely_integrable_component_lbound [OF absint_g])
by (simp add: gf)
qed
qed
qed
end