theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
begin
lemma le_left_mono: "x ≤ y ⟹ y ≤ a ⟶ x ≤ (a::'a::preorder)"
by (auto intro: order_trans)
lemma ball_trans:
assumes "y ∈ ball z q" "r + q ≤ s" shows "ball y r ⊆ ball z s"
proof safe
fix x assume x: "x ∈ ball y r"
have "dist z x ≤ dist z y + dist y x"
by (rule dist_triangle)
also have "… < s"
using assms x by auto
finally show "x ∈ ball z s"
by simp
qed
lemma has_integral_implies_lebesgue_measurable_cbox:
fixes f :: "'a :: euclidean_space ⇒ real"
assumes f: "(f has_integral I) (cbox x y)"
shows "f ∈ lebesgue_on (cbox x y) →⇩M borel"
proof (rule cld_measure.borel_measurable_cld)
let ?L = "lebesgue_on (cbox x y)"
let ?μ = "emeasure ?L"
let ?μ' = "outer_measure_of ?L"
interpret L: finite_measure ?L
proof
show "?μ (space ?L) ≠ ∞"
by (simp add: emeasure_restrict_space space_restrict_space emeasure_lborel_cbox_eq)
qed
show "cld_measure ?L"
proof
fix B A assume "B ⊆ A" "A ∈ null_sets ?L"
then show "B ∈ sets ?L"
using null_sets_completion_subset[OF ‹B ⊆ A›, of lborel]
by (auto simp add: null_sets_restrict_space sets_restrict_space_iff intro: )
next
fix A assume "A ⊆ space ?L" "⋀B. B ∈ sets ?L ⟹ ?μ B < ∞ ⟹ A ∩ B ∈ sets ?L"
from this(1) this(2)[of "space ?L"] show "A ∈ sets ?L"
by (auto simp: Int_absorb2 less_top[symmetric])
qed auto
then interpret cld_measure ?L
.
have content_eq_L: "A ∈ sets borel ⟹ A ⊆ cbox x y ⟹ content A = measure ?L A" for A
by (subst measure_restrict_space) (auto simp: measure_def)
fix E and a b :: real assume "E ∈ sets ?L" "a < b" "0 < ?μ E" "?μ E < ∞"
then obtain M :: real where "?μ E = M" "0 < M"
by (cases "?μ E") auto
define e where "e = M / (4 + 2 / (b - a))"
from ‹a < b› ‹0<M› have "0 < e"
by (auto intro!: divide_pos_pos simp: field_simps e_def)
have "e < M / (3 + 2 / (b - a))"
using ‹a < b› ‹0 < M›
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 ‹0<M› ‹0 < e› ‹a < b› by (simp add: field_simps)
have e_less_M: "e < M / 1"
unfolding e_def using ‹a < b› ‹0<M› by (intro divide_strict_left_mono) (auto simp: field_simps)
obtain d
where "gauge d"
and integral_f: "∀p. p tagged_division_of cbox x y ∧ d fine p ⟶
norm ((∑(x,k) ∈ p. content k *⇩R f x) - I) < e"
using ‹0<e› f unfolding has_integral by auto
define C where "C X m = X ∩ {x. ball x (1/Suc m) ⊆ 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 ⊆ space ?L" and eq: "?μ' X = ?μ E"
have "(SUP m. outer_measure_of ?L (C X m)) = outer_measure_of ?L (⋃m. C X m)"
using ‹X ⊆ space ?L› by (intro SUP_outer_measure_of_incseq ‹incseq (C X)›) (auto simp: C_def)
also have "(⋃m. C X m) = X"
proof -
{ fix x
obtain e where "0 < e" "ball x e ⊆ d x"
using gaugeD[OF ‹gauge d›, of x] unfolding open_contains_ball by auto
moreover
obtain n where "1 / (1 + real n) < e"
using reals_Archimedean[OF ‹0<e›] by (auto simp: inverse_eq_divide)
then have "ball x (1 / (1 + real n)) ⊆ ball x e"
by (intro subset_ball) auto
ultimately have "∃n. ball x (1 / (1 + real n)) ⊆ d x"
by blast }
then show ?thesis
by (auto simp: C_def)
qed
finally have "(SUP m. outer_measure_of ?L (C X m)) = ?μ E"
using eq by auto
also have "… > M - e"
using ‹0 < M› ‹?μ E = M› ‹0<e› by (auto intro!: ennreal_lessI)
finally have "∃m. M - e < outer_measure_of ?L (C X m)"
unfolding less_SUP_iff by auto }
note C = this
let ?E = "{x∈E. f x ≤ a}" and ?F = "{x∈E. b ≤ f x}"
have "¬ (?μ' ?E = ?μ E ∧ ?μ' ?F = ?μ E)"
proof
assume eq: "?μ' ?E = ?μ E ∧ ?μ' ?F = ?μ E"
with C[of ?E] C[of ?F] ‹E ∈ sets ?L›[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 ‹incseq (C ?E)›, of ma m, THEN outer_measure_of_mono]
incseqD[OF ‹incseq (C ?F)›, of mb m, THEN outer_measure_of_mono]
by (auto intro: less_le_trans)
define d' where "d' x = d x ∩ ball x (1 / (3 * Suc m))" for x
have "gauge d'"
unfolding d'_def by (intro gauge_Int ‹gauge d› 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 ∩ (C ?E m) ≠ {} ∧ k ∩ (C ?F m) ≠ {}}"
define T where "T E k = (SOME x. x ∈ k ∩ C E m)" for E k
let ?A = "(λ(x, k). (T ?E k, k)) ` (p ∩ s) ∪ (p - s)"
let ?B = "(λ(x, k). (T ?F k, k)) ` (p ∩ s) ∪ (p - s)"
{ fix X assume X_eq: "X = ?E ∨ X = ?F"
let ?T = "(λ(x, k). (T X k, k))"
let ?p = "?T ` (p ∩ s) ∪ (p - s)"
have in_s: "(x, k) ∈ s ⟹ T X k ∈ k ∩ C X m" for x k
using someI_ex[of "λx. x ∈ k ∩ C X m"] X_eq unfolding ex_in_conv by (auto simp: T_def s_def)
{ fix x k assume "(x, k) ∈ p" "(x, k) ∈ s"
have k: "k ⊆ ball x (1 / (3 * Suc m))"
using ‹d' fine p›[THEN fineD, OF ‹(x, k) ∈ p›] by (auto simp: d'_def)
then have "x ∈ ball (T X k) (1 / (3 * Suc m))"
using in_s[OF ‹(x, k) ∈ s›] by (auto simp: C_def subset_eq dist_commute)
then have "ball x (1 / (3 * Suc m)) ⊆ ball (T X k) (1 / Suc m)"
by (rule ball_trans) (auto simp: divide_simps)
with k in_s[OF ‹(x, k) ∈ s›] have "k ⊆ d (T X k)"
by (auto simp: C_def) }
then have "d fine ?p"
using ‹d fine p› 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) ∈ ?p"
then consider "(z, k) ∈ p" "(z, k) ∉ s"
| x' where "(x', k) ∈ p" "(x', k) ∈ s" "z = T X k"
by (auto simp: T_def)
then have "z ∈ k ∧ k ⊆ cbox x y ∧ (∃a b. k = cbox a b)"
using p(1) by cases (auto dest: in_s)
then show "z ∈ k" "k ⊆ cbox x y" "∃a b. k = cbox a b"
by auto
next
fix z k z' k' assume "(z, k) ∈ ?p" "(z', k') ∈ ?p" "(z, k) ≠ (z', k')"
with tagged_division_ofD(5)[OF p(1), of _ k _ k']
show "interior k ∩ interior k' = {}"
by (auto simp: T_def dest: in_s)
next
have "{k. ∃x. (x, k) ∈ ?p} = {k. ∃x. (x, k) ∈ p}"
by (auto simp: T_def image_iff Bex_def)
then show "⋃{k. ∃x. (x, k) ∈ ?p} = cbox x y"
using p(1) by auto
qed
ultimately have I: "norm ((∑(x,k) ∈ ?p. content k *⇩R f x) - I) < e"
using integral_f by auto
have "(∑(x,k) ∈ ?p. content k *⇩R f x) =
(∑(x,k) ∈ ?T ` (p ∩ s). content k *⇩R f x) + (∑(x,k) ∈ p - s. content k *⇩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 "(∑(x,k) ∈ ?T ` (p ∩ s). content k *⇩R f x) = (∑(x,k) ∈ p ∩ s. content k *⇩R f (T X k))"
proof (subst sum.reindex_nontrivial, safe)
fix x1 x2 k assume 1: "(x1, k) ∈ p" "(x1, k) ∈ s" and 2: "(x2, k) ∈ p" "(x2, k) ∈ s"
and eq: "content k *⇩R f (T X k) ≠ 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 ‹auto intro!: sum.cong›)
finally have eq: "(∑(x,k) ∈ ?p. content k *⇩R f x) =
(∑(x,k) ∈ p ∩ s. content k *⇩R f (T X k)) + (∑(x,k) ∈ p - s. content k *⇩R f x)" .
have in_T: "(x, k) ∈ s ⟹ T X k ∈ 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) ∈ p ⟹ k ∈ 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) ≤ (∑(x,k) ∈ p ∩ s. content k) * (b - a)"
proof (intro mult_right_mono)
have fin: "?μ (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}}) < ∞" for X
using ‹?μ E < ∞› by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono ‹E ∈ sets ?L›)
have sets: "(E ∩ ⋃{k∈snd`p. k ∩ C X m = {}}) ∈ sets ?L" for X
using tagged_division_ofD(1)[OF p(1)] by (intro sets.Diff ‹E ∈ sets ?L› sets.finite_Union sets.Int) (auto intro: p_in_L)
{ fix X assume "X ⊆ E" "M - e < ?μ' (C X m)"
have "M - e ≤ ?μ' (C X m)"
by (rule less_imp_le) fact
also have "… ≤ ?μ' (E - (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}}))"
proof (intro outer_measure_of_mono subsetI)
fix v assume "v ∈ C X m"
then have "v ∈ cbox x y" "v ∈ E"
using ‹E ⊆ space ?L› ‹X ⊆ E› by (auto simp: space_restrict_space C_def)
then obtain z k where "(z, k) ∈ p" "v ∈ k"
using tagged_division_ofD(6)[OF p(1), symmetric] by auto
then show "v ∈ E - E ∩ (⋃{k∈snd`p. k ∩ C X m = {}})"
using ‹v ∈ C X m› ‹v ∈ E› by auto
qed
also have "… = ?μ E - ?μ (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}})"
using ‹E ∈ sets ?L› fin[of X] sets[of X] by (auto intro!: emeasure_Diff)
finally have "?μ (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}}) ≤ e"
using ‹0 < e› e_less_M apply (cases "?μ (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}})")
by (auto simp add: ‹?μ E = M› ennreal_minus ennreal_le_iff2)
note this }
note upper_bound = this
have "?μ (E ∩ ⋃(snd`(p - s))) =
?μ ((E ∩ ⋃{k∈snd`p. k ∩ C ?E m = {}}) ∪ (E ∩ ⋃{k∈snd`p. k ∩ C ?F m = {}}))"
by (intro arg_cong[where f="?μ"]) (auto simp: s_def image_def Bex_def)
also have "… ≤ ?μ (E ∩ ⋃{k∈snd`p. k ∩ C ?E m = {}}) + ?μ (E ∩ ⋃{k∈snd`p. k ∩ C ?F m = {}})"
using sets[of ?E] sets[of ?F] M_minus_e by (intro emeasure_subadditive) auto
also have "… ≤ e + ennreal e"
using upper_bound[of ?E] upper_bound[of ?F] M_minus_e by (intro add_mono) auto
finally have "?μ E - 2*e ≤ ?μ (E - (E ∩ ⋃(snd`(p - s))))"
using ‹0 < e› ‹E ∈ sets ?L› tagged_division_ofD(1)[OF p(1)]
by (subst emeasure_Diff)
(auto simp: ennreal_plus[symmetric] top_unique simp del: ennreal_plus
intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
also have "… ≤ ?μ (⋃x∈p ∩ s. snd x)"
proof (safe intro!: emeasure_mono subsetI)
fix v assume "v ∈ E" and not: "v ∉ (⋃x∈p ∩ s. snd x)"
then have "v ∈ cbox x y"
using ‹E ⊆ space ?L› by (auto simp: space_restrict_space)
then obtain z k where "(z, k) ∈ p" "v ∈ k"
using tagged_division_ofD(6)[OF p(1), symmetric] by auto
with not show "v ∈ UNION (p - s) snd"
by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"])
qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
also have "… = measure ?L (⋃x∈p ∩ s. snd x)"
by (auto intro!: emeasure_eq_ennreal_measure)
finally have "M - 2 * e ≤ measure ?L (⋃x∈p ∩ s. snd x)"
unfolding ‹?μ E = M› using ‹0 < e› by (simp add: ennreal_minus)
also have "measure ?L (⋃x∈p ∩ s. snd x) = content (⋃x∈p ∩ 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 (⋃x∈p ∩ s. snd x) ≤ (∑k∈p ∩ 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 ≤ (∑(x, y)∈p ∩ s. content y)"
using ‹0 < e› by (simp add: split_beta)
qed (use ‹a < b› in auto)
also have "… = (∑(x,k) ∈ p ∩ s. content k * (b - a))"
by (simp add: sum_distrib_right split_beta')
also have "… ≤ (∑(x,k) ∈ p ∩ 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 "… = (∑(x,k) ∈ p ∩ s. content k * f (T ?F k)) - (∑(x,k) ∈ p ∩ s. content k * f (T ?E k))"
by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
also have "… = (∑(x,k) ∈ ?B. content k *⇩R f x) - (∑(x,k) ∈ ?A. content k *⇩R f x)"
by (subst (1 2) parts) auto
also have "… ≤ norm ((∑(x,k) ∈ ?B. content k *⇩R f x) - (∑(x,k) ∈ ?A. content k *⇩R f x))"
by auto
also have "… ≤ e + e"
using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
finally show False
using ‹2 * e < (b - a) * (M - e * 3)› by (auto simp: field_simps)
qed
moreover have "?μ' ?E ≤ ?μ E" "?μ' ?F ≤ ?μ E"
unfolding outer_measure_of_eq[OF ‹E ∈ sets ?L›, symmetric] by (auto intro!: outer_measure_of_mono)
ultimately show "min (?μ' ?E) (?μ' ?F) < ?μ E"
unfolding min_less_iff_disj by (auto simp: less_le)
qed
lemma has_integral_implies_lebesgue_measurable_real:
fixes f :: "'a :: euclidean_space ⇒ real"
assumes f: "(f has_integral I) Ω"
shows "(λx. f x * indicator Ω x) ∈ lebesgue →⇩M borel"
proof -
define B :: "nat ⇒ 'a set" where "B n = cbox (- real n *⇩R One) (real n *⇩R One)" for n
show "(λx. f x * indicator Ω x) ∈ lebesgue →⇩M borel"
proof (rule measurable_piecewise_restrict)
have "(⋃n. box (- real n *⇩R One) (real n *⇩R One)) ⊆ UNION UNIV B"
unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
then show "countable (range B)" "space lebesgue ⊆ UNION UNIV B"
by (auto simp: B_def UN_box_eq_UNIV)
next
fix Ω' assume "Ω' ∈ range B"
then obtain n where Ω': "Ω' = B n" by auto
then show "Ω' ∩ space lebesgue ∈ sets lebesgue"
by (auto simp: B_def)
have "f integrable_on Ω"
using f by auto
then have "(λx. f x * indicator Ω x) integrable_on Ω"
by (auto simp: integrable_on_def cong: has_integral_cong)
then have "(λx. f x * indicator Ω x) integrable_on (Ω ∪ B n)"
by (rule integrable_on_superset) auto
then have "(λx. f x * indicator Ω x) integrable_on B n"
unfolding B_def by (rule integrable_on_subcbox) auto
then show "(λx. f x * indicator Ω x) ∈ lebesgue_on Ω' →⇩M borel"
unfolding B_def Ω' 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 ⇒ 'b :: euclidean_space"
assumes f: "(f has_integral I) Ω"
shows "(λx. indicator Ω x *⇩R f x) ∈ lebesgue →⇩M borel"
proof (intro borel_measurable_euclidean_space[where 'c='b, THEN iffD2] ballI)
fix i :: "'b" assume "i ∈ Basis"
have "(λx. (f x ∙ i) * indicator Ω x) ∈ 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 "(λx. indicator Ω x *⇩R f x ∙ i) ∈ borel_measurable (completion lborel)"
by (simp add: ac_simps)
qed
subsection ‹Equivalence Lebesgue integral on @{const lborel} and HK-integral›
lemma has_integral_measure_lborel:
fixes A :: "'a::euclidean_space set"
assumes A[measurable]: "A ∈ sets borel" and finite: "emeasure lborel A < ∞"
shows "((λx. 1) has_integral measure lborel A) A"
proof -
{ fix l u :: 'a
have "((λx. 1) has_integral measure lborel (box l u)) (box l u)"
proof cases
assume "∀b∈Basis. l ∙ b ≤ u ∙ 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="λ_. 1"])
using has_integral_const[of "1::real" l u]
apply (simp_all add: inner_diff_left[symmetric] content_cbox_cases)
done
next
assume "¬ (∀b∈Basis. l ∙ b ≤ u ∙ 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 = "λA. measure lborel (A ∩ box a b)"
have "Int_stable (range (λ(a, b). box a b))"
by (auto simp: Int_stable_def box_Int_box)
moreover have "(range (λ(a, b). box a b)) ⊆ Pow UNIV"
by auto
moreover have "A ∈ sigma_sets UNIV (range (λ(a, b). box a b))"
using A unfolding borel_eq_box by simp
ultimately have "((λx. 1) has_integral ?M A) (A ∩ 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 ∈ sets borel"
by (simp add: borel_eq_box)
have "((λx. 1) has_integral ?M (box a b)) (box a b)"
by (simp add: has_integral_box)
moreover have "((λx. if x ∈ A ∩ box a b then 1 else 0) has_integral ?M A) (box a b)"
by (subst has_integral_restrict) (auto intro: compl)
ultimately have "((λx. 1 - (if x ∈ A ∩ 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 "((λx. (if x ∈ (UNIV - A) ∩ 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 "((λx. 1) has_integral ?M (box a b) - ?M A) ((UNIV - A) ∩ 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]: "⋀i. F i ∈ sets borel"
by (simp add: borel_eq_box subset_eq)
have "((λx. if x ∈ UNION UNIV F ∩ box a b then 1 else 0) has_integral ?M (⋃i. F i)) (box a b)"
proof (rule has_integral_monotone_convergence_increasing)
let ?f = "λk x. ∑i<k. if x ∈ F i ∩ box a b then 1 else 0 :: real"
show "⋀k. (?f k has_integral (∑i<k. ?M (F i))) (box a b)"
using union.IH by (auto intro!: has_integral_sum simp del: Int_iff)
show "⋀k x. ?f k x ≤ ?f (Suc k) x"
by (intro sum_mono2) auto
from union(1) have *: "⋀x i j. x ∈ F i ⟹ x ∈ F j ⟷ j = i"
by (auto simp add: disjoint_family_on_def)
show "⋀x. (λk. ?f k x) ⇢ (if x ∈ UNION UNIV F ∩ 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 ((⋃x. F x) ∩ box a b) ≤ emeasure lborel (box a b)"
by (intro emeasure_mono) auto
with union(1) show "(λk. ∑i<k. ?M (F i)) ⇢ ?M (⋃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 = "λn::nat. box (- real n *⇩R One) (real n *⇩R One) :: 'a set"
let ?f = "λn::nat. λx. if x ∈ A ∩ ?B n then 1 else 0 :: real"
let ?M = "λn. measure lborel (A ∩ ?B n)"
show "⋀n::nat. (?f n has_integral ?M n) A"
using * by (subst has_integral_restrict) simp_all
show "⋀k x. ?f k x ≤ ?f (Suc k) x"
by (auto simp: box_def)
{ fix x assume "x ∈ A"
moreover have "(λk. indicator (A ∩ ?B k) x :: real) ⇢ indicator (⋃k::nat. A ∩ ?B k) x"
by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def box_def)
ultimately show "(λk. if x ∈ A ∩ ?B k then 1 else 0::real) ⇢ 1"
by (simp add: indicator_def UN_box_eq_UNIV) }
have "(λn. emeasure lborel (A ∩ ?B n)) ⇢ emeasure lborel (⋃n::nat. A ∩ ?B n)"
by (intro Lim_emeasure_incseq) (auto simp: incseq_def box_def)
also have "(λn. emeasure lborel (A ∩ ?B n)) = (λn. measure lborel (A ∩ ?B n))"
proof (intro ext emeasure_eq_ennreal_measure)
fix n have "emeasure lborel (A ∩ ?B n) ≤ emeasure lborel (?B n)"
by (intro emeasure_mono) auto
then show "emeasure lborel (A ∩ ?B n) ≠ top"
by (auto simp: top_unique)
qed
finally show "(λn. measure lborel (A ∩ ?B n)) ⇢ 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 ⇒ real"
assumes f: "f ∈ borel_measurable borel" "⋀x. 0 ≤ f x" "(∫⇧+x. f x ∂lborel) = ennreal r" "0 ≤ r"
shows "(f has_integral r) UNIV"
using f proof (induct f arbitrary: r rule: borel_measurable_induct_real)
case (set A)
then have "((λx. 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 * (∫⇧+ x. g x ∂lborel) = ennreal r"
by (subst nn_integral_cmult[symmetric]) (auto simp: ennreal_mult)
with ‹0 ≤ r› ‹0 ≤ c›
obtain r' where "(c = 0 ∧ r = 0) ∨ (0 ≤ r' ∧ (∫⇧+ x. ennreal (g x) ∂lborel) = ennreal r' ∧ r = c * r')"
by (cases "∫⇧+ x. ennreal (g x) ∂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 "(∫⇧+ x. h x + g x ∂lborel) = (∫⇧+ x. h x ∂lborel) + (∫⇧+ x. g x ∂lborel)"
by (simp add: nn_integral_add)
with add obtain a b where "0 ≤ a" "0 ≤ b" "(∫⇧+ x. h x ∂lborel) = ennreal a" "(∫⇧+ x. g x ∂lborel) = ennreal b" "r = a + b"
by (cases "∫⇧+ x. h x ∂lborel" "∫⇧+ x. g x ∂lborel" rule: ennreal2_cases)
(auto simp: add_top nn_integral_add top_add ennreal_plus[symmetric] simp del: ennreal_plus)
with add show ?case
by (auto intro!: has_integral_add)
next
case (seq U)
note seq(1)[measurable] and f[measurable]
{ fix i x
have "U i x ≤ 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 "(∫⇧+x. U i x ∂lborel) ≤ (∫⇧+x. f x ∂lborel)"
using seq(2) f(2) U_le_f by (intro nn_integral_mono) simp
then obtain p where "(∫⇧+x. U i x ∂lborel) = ennreal p" "p ≤ r" "0 ≤ p"
using seq(6) ‹0≤r› by (cases "∫⇧+x. U i x ∂lborel" rule: ennreal_cases) (auto simp: top_unique)
moreover note seq
ultimately have "∃p. (∫⇧+x. U i x ∂lborel) = ennreal p ∧ 0 ≤ p ∧ p ≤ r ∧ (U i has_integral p) UNIV"
by auto }
then obtain p where p: "⋀i. (∫⇧+x. ennreal (U i x) ∂lborel) = ennreal (p i)"
and bnd: "⋀i. p i ≤ r" "⋀i. 0 ≤ p i"
and U_int: "⋀i.(U i has_integral (p i)) UNIV" by metis
have int_eq: "⋀i. integral UNIV (U i) = p i" using U_int by (rule integral_unique)
have *: "f integrable_on UNIV ∧ (λk. integral UNIV (U k)) ⇢ integral UNIV f"
proof (rule monotone_convergence_increasing)
show "⋀k. U k integrable_on UNIV" using U_int by auto
show "⋀k x. x∈UNIV ⟹ U k x ≤ U (Suc k) x" using ‹incseq U› by (auto simp: incseq_def le_fun_def)
then show "bounded (range (λk. integral UNIV (U k)))"
using bnd int_eq by (auto simp: bounded_real intro!: exI[of _ r])
show "⋀x. x∈UNIV ⟹ (λk. U k x) ⇢ f x"
using seq by auto
qed
moreover have "(λi. (∫⇧+x. U i x ∂lborel)) ⇢ (∫⇧+x. f x ∂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 ⇒ real"
assumes f: "f ∈ borel_measurable borel" "⋀x. 0 ≤ f x" "(∫⇧+x. f x ∂lborel) < ∞"
shows "(∫⇧+x. f x ∂lborel) = integral UNIV f"
proof -
from f(3) obtain r where r: "(∫⇧+x. f x ∂lborel) = ennreal r" "0 ≤ r"
by (cases "∫⇧+x. f x ∂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 ⇒ real"
assumes f: "f ∈ borel_measurable borel" "⋀x. 0 ≤ f x" "(∫⇧+x. f x ∂lborel) < ∞"
shows "f integrable_on UNIV"
proof -
from f(3) obtain r where r: "(∫⇧+x. f x ∂lborel) = ennreal r" "0 ≤ r"
by (cases "∫⇧+x. f x ∂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 ⇒ real"
assumes f_borel: "f ∈ borel_measurable borel" and nonneg: "⋀x. 0 ≤ f x"
assumes I: "(f has_integral I) UNIV"
shows "integral⇧N lborel f = I"
proof -
from f_borel have "(λx. ennreal (f x)) ∈ borel_measurable lborel" by auto
from borel_measurable_implies_simple_function_sequence'[OF this]
obtain F where F: "⋀i. simple_function lborel (F i)" "incseq F"
"⋀i x. F i x < top" "⋀x. (SUP i. F i x) = ennreal (f x)"
by blast
then have [measurable]: "⋀i. F i ∈ borel_measurable lborel"
by (metis borel_measurable_simple_function)
let ?B = "λi::nat. box (- (real i *⇩R One)) (real i *⇩R One) :: 'a set"
have "0 ≤ I"
using I by (rule has_integral_nonneg) (simp add: nonneg)
have F_le_f: "enn2real (F i x) ≤ f x" for i x
using F(3,4)[where x=x] nonneg SUP_upper[of i UNIV "λi. F i x"]
by (cases "F i x" rule: ennreal_cases) auto
let ?F = "λi x. F i x * indicator (?B i) x"
have "(∫⇧+ x. ennreal (f x) ∂lborel) = (SUP i. integral⇧N lborel (λx. ?F i x))"
proof (subst nn_integral_monotone_convergence_SUP[symmetric])
{ fix x
obtain j where j: "x ∈ ?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 "… = (SUP i. ?F i x)"
proof (rule SUP_eq)
fix i show "∃j∈UNIV. F i x ≤ ?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 "(∫⇧+ x. ennreal (f x) ∂lborel) = (∫⇧+ x. (SUP i. ?F i x) ∂lborel)"
by simp
qed (insert F, auto simp: incseq_def le_fun_def box_def split: split_indicator)
also have "… ≤ ennreal I"
proof (rule SUP_least)
fix i :: nat
have finite_F: "(∫⇧+ x. ennreal (enn2real (F i x) * indicator (?B i) x) ∂lborel) < ∞"
proof (rule nn_integral_bound_simple_function)
have "emeasure lborel {x ∈ space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) ≠ 0} ≤
emeasure lborel (?B i)"
by (intro emeasure_mono) (auto split: split_indicator)
then show "emeasure lborel {x ∈ space lborel. ennreal (enn2real (F i x) * indicator (?B i) x) ≠ 0} < ∞"
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: "(λx. 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 "(∫⇧+ x. F i x * indicator (?B i) x ∂lborel) =
(∫⇧+ x. ennreal (enn2real (F i x) * indicator (?B i) x) ∂lborel)"
using F(3,4)
by (intro nn_integral_cong) (auto simp: image_iff eq_commute split: split_indicator)
also have "… = ennreal (integral UNIV (λx. 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 "… ≤ ennreal I"
by (auto intro!: has_integral_le[OF integrable_integral[OF int_F] I] nonneg F_le_f
simp: ‹0 ≤ I› split: split_indicator )
finally show "(∫⇧+ x. F i x * indicator (?B i) x ∂lborel) ≤ ennreal I" .
qed
finally have "(∫⇧+ x. ennreal (f x) ∂lborel) < ∞"
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 ∈ sets borel" and [simp]: "0 ≤ r"
shows "((λx. 1) has_integral r) A ⟷ emeasure lborel A = ennreal r"
proof (cases "emeasure lborel A = ∞")
case emeasure_A: True
have "¬ (λx. 1::real) integrable_on A"
proof
assume int: "(λx. 1::real) integrable_on A"
then have "(indicator A::'a ⇒ real) integrable_on UNIV"
unfolding indicator_def[abs_def] integrable_restrict_UNIV .
then obtain r where "((indicator A::'a⇒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 "((λx. 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 ⇒ real"
assumes f: "integrable lborel f"
shows "(f has_integral (integral⇧L lborel f)) UNIV"
proof -
from integrableE[OF f] obtain r q
where "0 ≤ r" "0 ≤ q"
and r: "(∫⇧+ x. ennreal (max 0 (f x)) ∂lborel) = ennreal r"
and q: "(∫⇧+ x. ennreal (max 0 (- f x)) ∂lborel) = ennreal q"
and f: "f ∈ borel_measurable lborel" and eq: "integral⇧L lborel f = r - q"
unfolding ennreal_max_0 by auto
then have "((λx. max 0 (f x)) has_integral r) UNIV" "((λx. 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 "(λx. 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 ∈ Ω ⟶ f x = g x"
shows "(f has_integral x) Ω = (g has_integral x) Ω"
proof -
from ae obtain N
where N: "N ∈ sets borel" "emeasure lborel N = 0" "{x. ¬ (x ∈ Ω ⟶ f x = g x)} ⊆ N"
by (auto elim!: AE_E)
then have not_N: "AE x in lborel. x ∉ N"
by (simp add: AE_iff_measurable)
show ?thesis
proof (rule has_integral_spike_eq[symmetric])
show "⋀x. x∈Ω - N ⟹ f x = g x" using N(3) by auto
show "negligible N"
unfolding negligible_def
proof (intro allI)
fix a b :: "'a"
let ?F = "λx::'a. if x ∈ cbox a b then indicator N x else 0 :: real"
have "integrable lborel ?F = integrable lborel (λx::'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 ⇒ real"
assumes nonneg: "⋀x. 0 ≤ f x" and I: "(f has_integral I) Ω"
shows "integral⇧N lborel (λx. indicator Ω x * f x) = I"
proof -
from I have "(λx. indicator Ω x *⇩R f x) ∈ lebesgue →⇩M borel"
by (rule has_integral_implies_lebesgue_measurable)
then obtain f' :: "'a ⇒ real"
where [measurable]: "f' ∈ borel →⇩M borel" and eq: "AE x in lborel. indicator Ω x * f x = f' x"
by (auto dest: completion_ex_borel_measurable_real)
from I have "((λx. abs (indicator Ω x * f x)) has_integral I) UNIV"
using nonneg by (simp add: indicator_def if_distrib[of "λx. x * f y" for y] cong: if_cong)
also have "((λx. abs (indicator Ω x * f x)) has_integral I) UNIV ⟷ ((λx. abs (f' x)) has_integral I) UNIV"
using eq by (intro has_integral_AE) auto
finally have "integral⇧N lborel (λx. abs (f' x)) = I"
by (rule nn_integral_has_integral_lborel[rotated 2]) auto
also have "integral⇧N lborel (λx. abs (f' x)) = integral⇧N lborel (λx. abs (indicator Ω 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: "⋀x. 0 ≤ f x"
shows "(f has_integral r) UNIV ⟷ (f ∈ lebesgue →⇩M borel ∧ integral⇧N lebesgue f = r ∧ 0 ≤ r)" (is "?I = ?N")
proof
assume ?I
have "0 ≤ r"
using has_integral_nonneg[OF ‹?I›] f by auto
then show ?N
using nn_integral_has_integral_lebesgue[OF f ‹?I›]
has_integral_implies_lebesgue_measurable[OF ‹?I›]
by (auto simp: nn_integral_completion)
next
assume ?N
then obtain f' where f': "f' ∈ borel →⇩M borel" "AE x in lborel. f x = f' x"
by (auto dest: completion_ex_borel_measurable_real)
moreover have "(∫⇧+ x. ennreal ¦f' x¦ ∂lborel) = (∫⇧+ x. ennreal ¦f x¦ ∂lborel)"
using f' by (intro nn_integral_cong_AE) auto
moreover have "((λx. ¦f' x¦) has_integral r) UNIV ⟷ ((λx. ¦f x¦) has_integral r) UNIV"
using f' by (intro has_integral_AE) auto
moreover note nn_integral_has_integral[of "λx. ¦f' x¦" r] ‹?N›
ultimately show ?I
using f by (auto simp: nn_integral_completion)
qed
context
fixes f::"'a::euclidean_space ⇒ 'b::euclidean_space"
begin
lemma has_integral_integral_lborel:
assumes f: "integrable lborel f"
shows "(f has_integral (integral⇧L lborel f)) UNIV"
proof -
have "((λx. ∑b∈Basis. (f x ∙ b) *⇩R b) has_integral (∑b∈Basis. integral⇧L lborel (λx. f x ∙ b) *⇩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: "(λx. ∑b∈Basis. (f x ∙ b) *⇩R b) = f"
by (simp add: fun_eq_iff euclidean_representation)
also have "(∑b∈Basis. integral⇧L lborel (λx. f x ∙ b) *⇩R b) = integral⇧L lborel f"
using f by (subst (2) eq_f[symmetric]) simp
finally show ?thesis .
qed
lemma integrable_on_lborel: "integrable lborel f ⟹ f integrable_on UNIV"
using has_integral_integral_lborel by auto
lemma integral_lborel: "integrable lborel f ⟹ integral UNIV f = (∫x. f x ∂lborel)"
using has_integral_integral_lborel by auto
end
context
begin
private lemma has_integral_integral_lebesgue_real:
fixes f :: "'a::euclidean_space ⇒ real"
assumes f: "integrable lebesgue f"
shows "(f has_integral (integral⇧L lebesgue f)) UNIV"
proof -
obtain f' where f': "f' ∈ borel →⇩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 "(∫⇧+ x. ennreal (norm (f x)) ∂lborel) = (∫⇧+ x. ennreal (norm (f' x)) ∂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⇧L lebesgue f = integral⇧L lebesgue f'"
using f' f by (intro integral_cong_AE) (auto intro: AE_completion measurable_completion)
moreover have "integral⇧L lebesgue f' = integral⇧L lborel f'"
using f' by (simp add: integral_completion)
moreover have "(f' has_integral integral⇧L lborel f') UNIV ⟷ (f has_integral integral⇧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 ⇒ 'b::euclidean_space"
assumes f: "integrable lebesgue f"
shows "(f has_integral (integral⇧L lebesgue f)) UNIV"
proof -
have "((λx. ∑b∈Basis. (f x ∙ b) *⇩R b) has_integral (∑b∈Basis. integral⇧L lebesgue (λx. f x ∙ b) *⇩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: "(λx. ∑b∈Basis. (f x ∙ b) *⇩R b) = f"
by (simp add: fun_eq_iff euclidean_representation)
also have "(∑b∈Basis. integral⇧L lebesgue (λx. f x ∙ b) *⇩R b) = integral⇧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 ⇒ 'b::euclidean_space"
shows "integrable lebesgue f ⟹ f integrable_on UNIV"
using has_integral_integral_lebesgue by auto
lemma integral_lebesgue:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "integrable lebesgue f ⟹ integral UNIV f = (∫x. f x ∂lebesgue)"
using has_integral_integral_lebesgue by auto
end
subsection ‹Absolute integrability (this is the same as Lebesgue integrability)›
translations
"LBINT x. f" == "CONST lebesgue_integral CONST lborel (λx. f)"
translations
"LBINT x:A. f" == "CONST set_lebesgue_integral CONST lborel A (λx. f)"
lemma set_integral_reflect:
fixes S and f :: "real ⇒ 'a :: {banach, second_countable_topology}"
shows "(LBINT x : S. f x) = (LBINT x : {x. - x ∈ S}. f (- x))"
by (subst lborel_integral_real_affine[where c="-1" and t=0])
(auto intro!: Bochner_Integration.integral_cong split: split_indicator)
lemma borel_integrable_atLeastAtMost':
fixes f :: "real ⇒ 'a::{banach, second_countable_topology}"
assumes f: "continuous_on {a..b} f"
shows "set_integrable lborel {a..b} f" (is "integrable _ ?f")
by (intro borel_integrable_compact compact_Icc f)
lemma integral_FTC_atLeastAtMost:
fixes f :: "real ⇒ 'a :: euclidean_space"
assumes "a ≤ b"
and F: "⋀x. a ≤ x ⟹ x ≤ b ⟹ (F has_vector_derivative f x) (at x within {a .. b})"
and f: "continuous_on {a .. b} f"
shows "integral⇧L lborel (λx. indicator {a .. b} x *⇩R f x) = F b - F a"
proof -
let ?f = "λx. indicator {a .. b} x *⇩R f x"
have "(?f has_integral (∫x. ?f x ∂lborel)) UNIV"
using borel_integrable_atLeastAtMost'[OF f] by (rule has_integral_integral_lborel)
moreover
have "(f has_integral F b - F a) {a .. b}"
by (intro fundamental_theorem_of_calculus ballI assms) auto
then have "(?f has_integral F b - F a) {a .. b}"
by (subst has_integral_cong[where g=f]) auto
then have "(?f has_integral F b - F a) UNIV"
by (intro has_integral_on_superset[where T=UNIV and S="{a..b}"]) auto
ultimately show "integral⇧L lborel ?f = F b - F a"
by (rule has_integral_unique)
qed
lemma set_borel_integral_eq_integral:
fixes f :: "'a::euclidean_space ⇒ '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 = "λx. indicator S x *⇩R f x"
have "(?f has_integral LINT x : S | lborel. f x) UNIV"
by (rule has_integral_integral_lborel) fact
hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
apply (subst has_integral_restrict_UNIV [symmetric])
apply (rule has_integral_eq)
by auto
thus "f integrable_on S"
by (auto simp add: integrable_on_def)
with 1 have "(f has_integral (integral S f)) S"
by (intro integrable_integral, auto simp add: integrable_on_def)
thus "LINT x : S | lborel. f x = integral S f"
by (intro has_integral_unique [OF 1])
qed
lemma has_integral_set_lebesgue:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "set_integrable lebesgue S f"
shows "(f has_integral (LINT x:S|lebesgue. f x)) S"
using has_integral_integral_lebesgue[OF f]
by (simp_all add: indicator_def if_distrib[of "λx. x *⇩R f _"] has_integral_restrict_UNIV cong: if_cong)
lemma set_lebesgue_integral_eq_integral:
fixes f :: "'a::euclidean_space ⇒ '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 ∈ lmeasurable ⟷ ((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 ⇒ 'b::{banach, second_countable_topology}) ⇒ 'a set ⇒ bool"
(infixr "absolutely'_integrable'_on" 46)
where "f absolutely_integrable_on s ≡ set_integrable lebesgue s f"
lemma absolutely_integrable_on_def:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f absolutely_integrable_on s ⟷ f integrable_on s ∧ (λx. norm (f x)) integrable_on s"
proof safe
assume f: "f absolutely_integrable_on s"
then have nf: "integrable lebesgue (λx. norm (indicator s x *⇩R f x))"
by (intro integrable_norm)
note integrable_on_lebesgue[OF f] integrable_on_lebesgue[OF nf]
moreover have
"(λx. indicator s x *⇩R f x) = (λx. if x ∈ s then f x else 0)"
"(λx. norm (indicator s x *⇩R f x)) = (λx. if x ∈ s then norm (f x) else 0)"
by auto
ultimately show "f integrable_on s" "(λx. norm (f x)) integrable_on s"
by (simp_all add: integrable_restrict_UNIV)
next
assume f: "f integrable_on s" and nf: "(λx. norm (f x)) integrable_on s"
show "f absolutely_integrable_on s"
proof (rule integrableI_bounded)
show "(λx. indicator s x *⇩R f x) ∈ borel_measurable lebesgue"
using f has_integral_implies_lebesgue_measurable[of f _ s] by (auto simp: integrable_on_def)
show "(∫⇧+ x. ennreal (norm (indicator s x *⇩R f x)) ∂lebesgue) < ∞"
using nf nn_integral_has_integral_lebesgue[of "λx. norm (f x)" _ s]
by (auto simp: integrable_on_def nn_integral_completion)
qed
qed
lemma absolutely_integrable_on_null [intro]:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "content (cbox a b) = 0 ⟹ 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 ⇒ 'b :: euclidean_space"
shows "f absolutely_integrable_on box a b ⟷
f absolutely_integrable_on cbox a b"
by (auto simp: integrable_on_open_interval absolutely_integrable_on_def)
lemma absolutely_integrable_restrict_UNIV:
"(λx. if x ∈ s then f x else 0) absolutely_integrable_on UNIV ⟷ f absolutely_integrable_on s"
by (intro arg_cong2[where f=integrable]) auto
lemma absolutely_integrable_onI:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "f integrable_on s ⟹ (λx. norm (f x)) integrable_on s ⟹ f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def by auto
lemma nonnegative_absolutely_integrable_1:
fixes f :: "'a :: euclidean_space ⇒ real"
assumes f: "f integrable_on A" and "⋀x. x ∈ A ⟹ 0 ≤ f x"
shows "f absolutely_integrable_on A"
apply (rule absolutely_integrable_onI [OF f])
using assms by (simp add: integrable_eq)
lemma absolutely_integrable_on_iff_nonneg:
fixes f :: "'a :: euclidean_space ⇒ real"
assumes "⋀x. x ∈ S ⟹ 0 ≤ f x" shows "f absolutely_integrable_on S ⟷ f integrable_on S"
proof -
{ assume "f integrable_on S"
then have "(λx. if x ∈ S then f x else 0) integrable_on UNIV"
by (simp add: integrable_restrict_UNIV)
then have "(λx. if x ∈ S then f x else 0) absolutely_integrable_on UNIV"
using ‹f integrable_on S› absolutely_integrable_restrict_UNIV assms nonnegative_absolutely_integrable_1 by blast
then have "f absolutely_integrable_on S"
using absolutely_integrable_restrict_UNIV by blast
}
then show ?thesis
unfolding absolutely_integrable_on_def by auto
qed
lemma lmeasurable_iff_integrable_on: "S ∈ lmeasurable ⟷ (λx. 1::real) integrable_on S"
by (subst absolutely_integrable_on_iff_nonneg[symmetric])
(simp_all add: lmeasurable_iff_integrable)
lemma lmeasure_integral_UNIV: "S ∈ lmeasurable ⟹ measure lebesgue S = integral UNIV (indicator S)"
by (simp add: lmeasurable_iff_has_integral integral_unique)
lemma lmeasure_integral: "S ∈ lmeasurable ⟹ measure lebesgue S = integral S (λx. 1::real)"
by (auto simp add: lmeasure_integral_UNIV indicator_def[abs_def] lmeasurable_iff_integrable_on)
lemma
assumes 𝒟: "𝒟 division_of S"
shows lmeasurable_division: "S ∈ lmeasurable" (is ?l)
and content_division: "(∑k∈𝒟. measure lebesgue k) = measure lebesgue S" (is ?m)
proof -
{ fix d1 d2 assume *: "d1 ∈ 𝒟" "d2 ∈ 𝒟" "d1 ≠ 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 ∈ sets lborel" "d2 ∈ sets lborel" "d1 ∩ d2 ⊆ (cbox a b - box a b) ∪ (cbox c d - box c d)"
by auto
moreover have "(cbox a b - box a b) ∪ (cbox c d - box c d) ∈ null_sets lborel"
by (intro null_sets.Un null_sets_cbox_Diff_box)
ultimately have "d1 ∩ d2 ∈ null_sets lborel"
by (blast intro: null_sets_subset) }
then show ?l ?m
unfolding division_ofD(6)[OF 𝒟, symmetric]
using division_ofD(1,4)[OF 𝒟]
by (auto intro!: measure_Union_AE[symmetric] simp: completion.AE_iff_null_sets Int_def[symmetric] pairwise_def null_sets_def)
qed
text ‹This should be an abbreviation for negligible.›
lemma negligible_iff_null_sets: "negligible S ⟷ S ∈ null_sets lebesgue"
proof
assume "negligible S"
then have "(indicator S has_integral (0::real)) UNIV"
by (auto simp: negligible)
then show "S ∈ 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 ∈ 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 "(λx. if x ∈ cbox a b then indicator S x else 0) ∈ lebesgue →⇩M borel"
using S by (auto intro!: measurable_If)
then show "(∫⇧+ x. ennreal (if x ∈ cbox a b then indicator S x else 0) ∂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: "⋀c x. ⟦(a + c *⇩R x) ∈ S; 0 ≤ c; a + x ∈ S⟧ ⟹ c = 1"
shows "negligible S"
proof -
have "negligible (op + (-a) ` S)"
proof (subst negligible_on_intervals, intro allI)
fix u v
show "negligible (op + (- a) ` S ∩ 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: "⋀c x. ⟦0 ≤ c; c < 1; a+x ∈ S⟧ ⟹ a + c *⇩R x ∉ S"
shows "negligible S"
proof -
show ?thesis
proof (rule starlike_negligible [OF ‹closed S›, of a])
fix c x
assume cx: "a + c *⇩R x ∈ S" "0 ≤ c" "a + x ∈ S"
with star have "~ (c < 1)" by auto
moreover have "~ (c > 1)"
using star [of "1/c" "c *⇩R x"] cx by force
ultimately show "c = 1" by arith
qed
qed
subsection‹Applications›
lemma negligible_hyperplane:
assumes "a ≠ 0 ∨ b ≠ 0" shows "negligible {x. a ∙ x = b}"
proof -
obtain x where x: "a ∙ x ≠ 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 ≠ 0" and a: "span S ⊆ {x. a ∙ x = 0}"
using lowdim_subset_hyperplane [OF assms] by blast
have "negligible (span S)"
using ‹a ≠ 0› a negligible_hyperplane by (blast intro: negligible_subset)
then show ?thesis
using span_inc by (blast intro: negligible_subset)
qed
proposition negligible_convex_frontier:
fixes S :: "'N :: euclidean_space set"
assumes "convex S"
shows "negligible(frontier S)"
proof -
have nf: "negligible(frontier S)" if "convex S" "0 ∈ S" for S :: "'N set"
proof -
obtain B where "B ⊆ S" and indB: "independent B"
and spanB: "S ⊆ span B" and cardB: "card B = dim S"
by (metis basis_exists)
consider "dim S < DIM('N)" | "dim S = DIM('N)"
using dim_subset_UNIV le_eq_less_or_eq by blast
then show ?thesis
proof cases
case 1
show ?thesis
by (rule negligible_subset [of "closure S"])
(simp_all add: Diff_subset frontier_def negligible_lowdim 1)
next
case 2
obtain a where a: "a ∈ interior S"
apply (rule interior_simplex_nonempty [OF indB])
apply (simp add: indB independent_finite)
apply (simp add: cardB 2)
apply (metis ‹B ⊆ S› ‹0 ∈ S› ‹convex S› 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 *⇩R x = (a + x) - (1 - c) *⇩R ((a + x) - a)"
by (simp add: algebra_simps)
assume "0 ≤ c" "c < 1" "a + x ∈ frontier S"
then show "a + c *⇩R x ∉ frontier S"
apply (clarsimp simp: frontier_def)
apply (subst eq)
apply (rule mem_interior_closure_convex_shrink [OF ‹convex S› 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 ∈ S" by auto
show ?thesis
using nf [of "(λx. -a + x) ` S"]
by (metis ‹a ∈ S› 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]: "¬ negligible UNIV"
unfolding negligible_iff_null_sets by (auto simp: null_sets_def emeasure_lborel_UNIV)
lemma negligible_interval:
"negligible (cbox a b) ⟷ box a b = {}" "negligible (box a b) ⟷ box a b = {}"
by (auto simp: negligible_iff_null_sets null_sets_def prod_nonneg inner_diff_left box_eq_empty
not_le emeasure_lborel_cbox_eq emeasure_lborel_box_eq
intro: eq_refl antisym less_imp_le)
subsection ‹Negligibility of a Lipschitz image of a negligible set›
lemma measure_eq_0_null_sets: "S ∈ null_sets M ⟹ measure M S = 0"
by (auto simp: measure_def null_sets_def)
text‹The bound will be eliminated by a sort of onion argument›
lemma locally_Lipschitz_negl_bounded:
fixes f :: "'M::euclidean_space ⇒ 'N::euclidean_space"
assumes MleN: "DIM('M) ≤ DIM('N)" "0 < B" "bounded S" "negligible S"
and lips: "⋀x. x ∈ S
⟹ ∃T. open T ∧ x ∈ T ∧
(∀y ∈ S ∩ T. norm(f y - f x) ≤ 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 ∈ lmeasurable"
using ‹negligible S› by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
using ‹0 < e› ‹0 < B› by (simp add: divide_simps)
obtain T
where "open T" "S ⊆ T" "T ∈ lmeasurable"
and "measure lebesgue T ≤ measure lebesgue S + e/2 / (2 * B * DIM('M)) ^ DIM('N)"
by (rule lmeasurable_outer_open [OF ‹S ∈ lmeasurable› e22])
then have T: "measure lebesgue T ≤ e/2 / (2 * B * DIM('M)) ^ DIM('N)"
using ‹negligible S› by (simp add: negligible_iff_null_sets measure_eq_0_null_sets)
have "∃r. 0 < r ∧ r ≤ 1/2 ∧
(x ∈ S ⟶ (∀y. norm(y - x) < r
⟶ y ∈ T ∧ (y ∈ S ⟶ norm(f y - f x) ≤ B * norm(y - x))))"
for x
proof (cases "x ∈ S")
case True
obtain U where "open U" "x ∈ U" and U: "⋀y. y ∈ S ∩ U ⟹ norm(f y - f x) ≤ B * norm(y - x)"
using lips [OF ‹x ∈ S›] by auto
have "x ∈ T ∩ U"
using ‹S ⊆ T› ‹x ∈ U› ‹x ∈ S› by auto
then obtain ε where "0 < ε" "ball x ε ⊆ T ∩ U"
by (metis ‹open T› ‹open U› openE open_Int)
then show ?thesis
apply (rule_tac x="min (1/2) ε" 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: "⋀x. 0 < R x ∧ R x ≤ 1/2"
and RT: "⋀x y. ⟦x ∈ S; norm(y - x) < R x⟧ ⟹ y ∈ T"
and RB: "⋀x y. ⟦x ∈ S; y ∈ S; norm(y - x) < R x⟧ ⟹ norm(f y - f x) ≤ B * norm(y - x)"
by metis+
then have gaugeR: "gauge (λx. ball x (R x))"
by (simp add: gauge_def)
obtain c where c: "S ⊆ cbox (-c *⇩R One) (c *⇩R One)" "box (-c *⇩R One:: 'M) (c *⇩R One) ≠ {}"
proof -
obtain B where B: "⋀x. x ∈ S ⟹ norm x ≤ B"
using ‹bounded S› 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: "⋃𝒟 ⊆ cbox (-c *⇩R One) (c *⇩R One)"
and cbox: "⋀K. K ∈ 𝒟 ⟹ interior K ≠ {} ∧ (∃c d. K = cbox c d)"
and pw: "pairwise (λA B. interior A ∩ interior B = {}) 𝒟"
and Ksub: "⋀K. K ∈ 𝒟 ⟹ ∃x ∈ S ∩ K. K ⊆ (λx. ball x (R x)) x"
and exN: "⋀u v. cbox u v ∈ 𝒟 ⟹ ∃n. ∀i ∈ Basis. v ∙ i - u ∙ i = (2*c) / 2^n"
and "S ⊆ ⋃𝒟"
using covering_lemma [OF c gaugeR] by force
have "∃u v z. K = cbox u v ∧ box u v ≠ {} ∧ z ∈ S ∧ z ∈ cbox u v ∧
cbox u v ⊆ ball z (R z)" if "K ∈ 𝒟" for K
proof -
obtain u v where "K = cbox u v"
using ‹K ∈ 𝒟› 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: "⋀K. K ∈ 𝒟 ⟹
K = cbox (uf K) (vf K) ∧ box (uf K) (vf K) ≠ {} ∧ zf K ∈ S ∧
zf K ∈ cbox (uf K) (vf K) ∧ cbox (uf K) (vf K) ⊆ ball (zf K) (R (zf K))"
by metis
define prj1 where "prj1 ≡ λx::'M. x ∙ (SOME i. i ∈ Basis)"
define fbx where "fbx ≡ λD. cbox (f(zf D) - (B * DIM('M) * (prj1(vf D - uf D))) *⇩R One::'N)
(f(zf D) + (B * DIM('M) * prj1(vf D - uf D)) *⇩R One)"
have vu_pos: "0 < prj1 (vf X - uf X)" if "X ∈ 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) ∙ i" if "X ∈ 𝒟" "i ∈ Basis" for X i
proof -
have "cbox (uf X) (vf X) ∈ 𝒟"
using uvz ‹X ∈ 𝒟› by auto
with exN obtain n where "⋀i. i ∈ Basis ⟹ vf X ∙ i - uf X ∙ i = (2*c) / 2^n"
by blast
then show ?thesis
by (simp add: ‹i ∈ Basis› SOME_Basis inner_diff prj1_def)
qed
have countbl: "countable (fbx ` 𝒟)"
using ‹countable 𝒟› by blast
have "(∑k∈fbx`𝒟'. measure lebesgue k) ≤ e/2" if "𝒟' ⊆ 𝒟" "finite 𝒟'" for 𝒟'
proof -
have BM_ge0: "0 ≤ B * (DIM('M) * prj1 (vf X - uf X))" if "X ∈ 𝒟'" for X
using ‹0 < B› ‹𝒟' ⊆ 𝒟› that vu_pos by fastforce
have "{} ∉ 𝒟'"
using cbox ‹𝒟' ⊆ 𝒟› interior_empty by blast
have "(∑k∈fbx`𝒟'. measure lebesgue k) ≤ sum (measure lebesgue o fbx) 𝒟'"
by (rule sum_image_le [OF ‹finite 𝒟'›]) (force simp: fbx_def)
also have "… ≤ (∑X∈𝒟'. (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X)"
proof (rule sum_mono)
fix X assume "X ∈ 𝒟'"
then have "X ∈ 𝒟" using ‹𝒟' ⊆ 𝒟› by blast
then have ufvf: "cbox (uf X) (vf X) = X"
using uvz by blast
have "prj1 (vf X - uf X) ^ DIM('M) = (∏i::'M ∈ Basis. prj1 (vf X - uf X))"
by (rule prod_constant [symmetric])
also have "… = (∏i∈Basis. vf X ∙ i - uf X ∙ i)"
using prj1_idem [OF ‹X ∈ 𝒟›] by (auto simp: algebra_simps intro: prod.cong)
finally have prj1_eq: "prj1 (vf X - uf X) ^ DIM('M) = (∏i∈Basis. vf X ∙ i - uf X ∙ i)" .
have "uf X ∈ cbox (uf X) (vf X)" "vf X ∈ cbox (uf X) (vf X)"
using uvz [OF ‹X ∈ 𝒟›] by (force simp: mem_box)+
moreover have "cbox (uf X) (vf X) ⊆ ball (zf X) (1/2)"
by (meson R12 order_trans subset_ball uvz [OF ‹X ∈ 𝒟›])
ultimately have "uf X ∈ ball (zf X) (1/2)" "vf X ∈ ball (zf X) (1/2)"
by auto
then have "dist (vf X) (uf X) ≤ 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) ≤ 1"
unfolding prj1_def dist_norm using Basis_le_norm SOME_Basis order_trans by fastforce
have 0: "0 ≤ prj1 (vf X - uf X)"
using ‹X ∈ 𝒟› prj1_def vu_pos by fastforce
have "(measure lebesgue ∘ fbx) X ≤ (2 * B * DIM('M)) ^ DIM('N) * content (cbox (uf X) (vf X))"
apply (simp add: fbx_def content_cbox_cases algebra_simps BM_ge0 ‹X ∈ 𝒟'› prod_constant)
apply (simp add: power_mult_distrib ‹0 < B› prj1_eq [symmetric])
using MleN 0 1 uvz ‹X ∈ 𝒟›
apply (fastforce simp add: box_ne_empty power_decreasing)
done
also have "… = (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X"
by (subst (3) ufvf[symmetric]) simp
finally show "(measure lebesgue ∘ fbx) X ≤ (2 * B * DIM('M)) ^ DIM('N) * measure lebesgue X" .
qed
also have "… = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) 𝒟'"
by (simp add: sum_distrib_left)
also have "… ≤ e/2"
proof -
have div: "𝒟' division_of ⋃𝒟'"
apply (auto simp: ‹finite 𝒟'› ‹{} ∉ 𝒟'› division_of_def)
using cbox that apply blast
using pairwise_subset [OF pw ‹𝒟' ⊆ 𝒟›] unfolding pairwise_def apply force+
done
have le_meaT: "measure lebesgue (⋃𝒟') ≤ measure lebesgue T"
proof (rule measure_mono_fmeasurable [OF _ _ ‹T : lmeasurable›])
show "(⋃𝒟') ∈ sets lebesgue"
using div lmeasurable_division by auto
have "⋃𝒟' ⊆ ⋃𝒟"
using ‹𝒟' ⊆ 𝒟› by blast
also have "... ⊆ T"
proof (clarify)
fix x D
assume "x ∈ D" "D ∈ 𝒟"
show "x ∈ T"
using Ksub [OF ‹D ∈ 𝒟›]
by (metis ‹x ∈ D› Int_iff dist_norm mem_ball norm_minus_commute subsetD RT)
qed
finally show "⋃𝒟' ⊆ T" .
qed
have "sum (measure lebesgue) 𝒟' = sum content 𝒟'"
using ‹𝒟' ⊆ 𝒟› 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 (⋃𝒟')"
using content_division [OF div] by auto
also have "… ≤ (2 * B * real DIM('M)) ^ DIM('N) * measure lebesgue T"
apply (rule mult_left_mono [OF le_meaT])
using ‹0 < B›
apply (simp add: algebra_simps)
done
also have "… ≤ e/2"
using T ‹0 < B› by (simp add: field_simps)
finally show ?thesis .
qed
finally show ?thesis .
qed
then have e2: "sum (measure lebesgue) 𝒢 ≤ e/2" if "𝒢 ⊆ fbx ` 𝒟" "finite 𝒢" for 𝒢
by (metis finite_subset_image that)
show "∃W∈lmeasurable. f ` S ⊆ W ∧ measure lebesgue W < e"
proof (intro bexI conjI)
have "∃X∈𝒟. f y ∈ fbx X" if "y ∈ S" for y
proof -
obtain X where "y ∈ X" "X ∈ 𝒟"
using ‹S ⊆ ⋃𝒟› ‹y ∈ S› by auto
then have y: "y ∈ ball(zf X) (R(zf X))"
using uvz by fastforce
have conj_le_eq: "z - b ≤ y ∧ y ≤ z + b ⟷ abs(y - z) ≤ b" for z y b::real
by auto
have yin: "y ∈ cbox (uf X) (vf X)" and zin: "(zf X) ∈ cbox (uf X) (vf X)"
using uvz ‹X ∈ 𝒟› ‹y ∈ X› by auto
have "norm (y - zf X) ≤ (∑i∈Basis. ¦(y - zf X) ∙ i¦)"
by (rule norm_le_l1)
also have "… ≤ real DIM('M) * prj1 (vf X - uf X)"
proof (rule sum_bounded_above)
fix j::'M assume j: "j ∈ Basis"
show "¦(y - zf X) ∙ j¦ ≤ prj1 (vf X - uf X)"
using yin zin j
by (fastforce simp add: mem_box prj1_idem [OF ‹X ∈ 𝒟› j] inner_diff_left)
qed
finally have nole: "norm (y - zf X) ≤ DIM('M) * prj1 (vf X - uf X)"
by simp
have fle: "¦f y ∙ i - f(zf X) ∙ i¦ ≤ B * DIM('M) * prj1 (vf X - uf X)" if "i ∈ Basis" for i
proof -
have "¦f y ∙ i - f (zf X) ∙ i¦ = ¦(f y - f (zf X)) ∙ i¦"
by (simp add: algebra_simps)
also have "… ≤ norm (f y - f (zf X))"
by (simp add: Basis_le_norm that)
also have "… ≤ B * norm(y - zf X)"
by (metis uvz RB ‹X ∈ 𝒟› dist_commute dist_norm mem_ball ‹y ∈ S› y)
also have "… ≤ B * real DIM('M) * prj1 (vf X - uf X)"
using ‹0 < B› 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 ‹X ∈ 𝒟›)
qed
then show "f ` S ⊆ (⋃D∈𝒟. fbx D)" by auto
next
have 1: "⋀D. D ∈ 𝒟 ⟹ fbx D ∈ lmeasurable"
by (auto simp: fbx_def)
have 2: "I' ⊆ 𝒟 ⟹ finite I' ⟹ measure lebesgue (⋃D∈I'. fbx D) ≤ e/2" for I'
by (rule order_trans[OF measure_Union_le e2]) (auto simp: fbx_def)
have 3: "0 ≤ e/2"
using ‹0<e› by auto
show "(⋃D∈𝒟. fbx D) ∈ lmeasurable"
by (intro fmeasurable_UN_bound[OF ‹countable 𝒟› 1 2 3])
have "measure lebesgue (⋃D∈𝒟. fbx D) ≤ e/2"
by (intro measure_UN_bound[OF ‹countable 𝒟› 1 2 3])
then show "measure lebesgue (⋃D∈𝒟. fbx D) < e"
using ‹0 < e› by linarith
qed
qed
proposition negligible_locally_Lipschitz_image:
fixes f :: "'M::euclidean_space ⇒ 'N::euclidean_space"
assumes MleN: "DIM('M) ≤ DIM('N)" "negligible S"
and lips: "⋀x. x ∈ S
⟹ ∃T B. open T ∧ x ∈ T ∧
(∀y ∈ S ∩ T. norm(f y - f x) ≤ B * norm(y - x))"
shows "negligible (f ` S)"
proof -
let ?S = "λn. ({x ∈ S. norm x ≤ n ∧
(∃T. open T ∧ x ∈ T ∧
(∀y∈S ∩ T. norm (f y - f x) ≤ (real n + 1) * norm (y - x)))})"
have negfn: "f ` ?S n ∈ 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 ‹negligible S›])
have "S = (⋃n. ?S n)"
proof (intro set_eqI iffI)
fix x assume "x ∈ S"
with lips obtain T B where T: "open T" "x ∈ T"
and B: "⋀y. y ∈ S ∩ T ⟹ norm(f y - f x) ≤ B * norm(y - x)"
by metis+
have no: "norm (f y - f x) ≤ (nat ⌈max B (norm x)⌉ + 1) * norm (y - x)" if "y ∈ S ∩ T" for y
proof -
have "B * norm(y - x) ≤ (nat ⌈max B (norm x)⌉ + 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 ∈ ?S (nat (ceiling (max B (norm x))))"
apply (simp add: ‹x ∈ S ›, rule)
using real_nat_ceiling_ge max.bounded_iff apply blast
using T no
apply (force simp: algebra_simps)
done
then show "x ∈ (⋃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 ⇒ 'N::euclidean_space"
assumes MleN: "DIM('M) ≤ DIM('N)" "negligible S"
and diff_f: "f differentiable_on S"
shows "negligible (f ` S)"
proof -
have "∃T B. open T ∧ x ∈ T ∧ (∀y ∈ S ∩ T. norm(f y - f x) ≤ B * norm(y - x))"
if "x ∈ S" for x
proof -
obtain f' where "linear f'"
and f': "⋀e. e>0 ⟹
∃d>0. ∀y∈S. norm (y - x) < d ⟶
norm (f y - f x - f' (y - x)) ≤ e * norm (y - x)"
using diff_f ‹x ∈ S›
by (auto simp: linear_linear differentiable_on_def differentiable_def has_derivative_within_alt)
obtain B where "B > 0" and B: "∀x. norm (f' x) ≤ B * norm x"
using linear_bounded_pos ‹linear f'› by blast
obtain d where "d>0"
and d: "⋀y. ⟦y ∈ S; norm (y - x) < d⟧ ⟹
norm (f y - f x - f' (y - x)) ≤ norm (y - x)"
using f' [of 1] by (force simp:)
have *: "norm (f y - f x) ≤ (B + 1) * norm (y - x)"
if "y ∈ S" "norm (y - x) < d" for y
proof -
have "norm (f y - f x) -B * norm (y - x) ≤ norm (f y - f x) - norm (f' (y - x))"
by (simp add: B)
also have "… ≤ norm (f y - f x - f' (y - x))"
by (rule norm_triangle_ineq2)
also have "... ≤ 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 ‹d>0›
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 ⇒ 'N::euclidean_space"
assumes MlessN: "DIM('M) < DIM('N)" and diff_f: "f differentiable_on S"
shows "negligible (f ` S)"
proof -
have "x ≤ DIM('M) ⟹ x ≤ DIM('N)" for x
using MlessN by linarith
obtain lift :: "'M * real ⇒ 'N" and drop :: "'N ⇒ 'M * real" and j :: 'N
where "linear lift" "linear drop" and dropl [simp]: "⋀z. drop (lift z) = z"
and "j ∈ Basis" and j: "⋀x. lift(x,0) ∙ j = 0"
using lowerdim_embeddings [OF MlessN] by metis
have "negligible {x. x∙j = 0}"
by (metis ‹j ∈ Basis› negligible_standard_hyperplane)
then have neg0S: "negligible ((λx. lift (x, 0)) ` S)"
apply (rule negligible_subset)
by (simp add: image_subsetI j)
have diff_f': "f ∘ fst ∘ drop differentiable_on (λx. lift (x, 0)) ` S"
using diff_f
apply (clarsimp simp add: differentiable_on_def)
apply (intro differentiable_chain_within linear_imp_differentiable [OF ‹linear drop›]
linear_imp_differentiable [OF fst_linear])
apply (force simp: image_comp o_def)
done
have "f = (f o fst o drop o (λx. lift (x, 0)))"
by (simp add: o_def)
then show ?thesis
apply (rule ssubst)
apply (subst image_comp [symmetric])
apply (metis negligible_differentiable_image_negligible order_refl diff_f' neg0S)
done
qed
lemma set_integral_norm_bound:
fixes f :: "_ ⇒ 'a :: {banach, second_countable_topology}"
shows "set_integrable M k f ⟹ norm (LINT x:k|M. f x) ≤ LINT x:k|M. norm (f x)"
using integral_norm_bound[of M "λx. indicator k x *⇩R f x"] by simp
lemma set_integral_finite_UN_AE:
fixes f :: "_ ⇒ _ :: {banach, second_countable_topology}"
assumes "finite I"
and ae: "⋀i j. i ∈ I ⟹ j ∈ I ⟹ AE x in M. (x ∈ A i ∧ x ∈ A j) ⟶ i = j"
and [measurable]: "⋀i. i ∈ I ⟹ A i ∈ sets M"
and f: "⋀i. i ∈ I ⟹ set_integrable M (A i) f"
shows "LINT x:(⋃i∈I. A i)|M. f x = (∑i∈I. LINT x:A i|M. f x)"
using ‹finite I› order_refl[of I]
proof (induction I rule: finite_subset_induct')
case (insert i I')
have "AE x in M. (∀j∈I'. x ∈ A i ⟶ x ∉ A j)"
proof (intro AE_ball_countable[THEN iffD2] ballI)
fix j assume "j ∈ I'"
with ‹I' ⊆ I› ‹i ∉ I'› have "i ≠ j" "j ∈ I"
by auto
then show "AE x in M. x ∈ A i ⟶ x ∉ A j"
using ae[of i j] ‹i ∈ I› by auto
qed (use ‹finite I'› in ‹rule countable_finite›)
then have "AE x∈A i in M. ∀xa∈I'. x ∉ A xa "
by auto
with insert.hyps insert.IH[symmetric]
show ?case
by (auto intro!: set_integral_Un_AE sets.finite_UN f set_integrable_UN)
qed simp
lemma set_integrable_norm:
fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}"
assumes f: "set_integrable M k f" shows "set_integrable M k (λx. norm (f x))"
using integrable_norm[OF f] by simp
lemma absolutely_integrable_bounded_variation:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "f absolutely_integrable_on UNIV"
obtains B where "∀d. d division_of (⋃d) ⟶ sum (λk. norm (integral k f)) d ≤ B"
proof (rule that[of "integral UNIV (λx. norm (f x))"]; safe)
fix d :: "'a set set" assume d: "d division_of ⋃d"
have *: "k ∈ d ⟹ 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 "(∑k∈d. norm (integral k f)) = (∑k∈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 "… ≤ (∑k∈d. LINT x:k|lebesgue. norm (f x))"
by (intro sum_mono set_integral_norm_bound *)
also have "… = (∑k∈d. integral k (λx. norm (f x)))"
by (intro sum.cong refl set_lebesgue_integral_eq_integral(2) set_integrable_norm *)
also have "… ≤ integral (⋃d) (λx. 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 "… ≤ integral UNIV (λx. norm (f x))"
using integrable_on_subdivision[OF d] assms unfolding absolutely_integrable_on_def
by (intro integral_subset_le) auto
finally show "(∑k∈d. norm (integral k f)) ≤ integral UNIV (λx. norm (f x))" .
qed
lemma absdiff_norm_less:
assumes "sum (λx. norm (f x - g x)) s < e"
and "finite s"
shows "¦sum (λx. norm(f x)) s - sum (λx. norm(g x)) s¦ < 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 ⇒ 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
and *: "⋀d. d division_of (cbox a b) ⟹ sum (λK. norm(integral K f)) d ≤ B"
shows "f absolutely_integrable_on cbox a b"
proof -
let ?f = "λd. ∑K∈d. norm (integral K f)" and ?D = "{d. d division_of (cbox a b)}"
have D_1: "?D ≠ {}"
by (rule elementary_interval[of a b]) auto
have D_2: "bdd_above (?f`?D)"
by (metis * mem_Collect_eq bdd_aboveI2)
note D = D_1 D_2
let ?S = "SUP x:?D. ?f x"
have *: "∃γ. gauge γ ∧
(∀p. p tagged_division_of cbox a b ∧
γ fine p ⟶
norm ((∑(x,k) ∈ p. content k *⇩R norm (f x)) - ?S) < e)"
if e: "e > 0" for e
proof -
have "?S - e/2 < ?S" using ‹e > 0› by simp
then obtain d where d: "d division_of (cbox a b)" "?S - e/2 < (∑k∈d. norm (integral k f))"
unfolding less_cSUP_iff[OF D] by auto
note d' = division_ofD[OF this(1)]
have "∃e>0. ∀i∈d. x ∉ i ⟶ ball x e ∩ i = {}" for x
proof -
have "∃d'>0. ∀x'∈⋃{i ∈ d. x ∉ i}. d' ≤ dist x x'"
proof (rule separate_point_closed)
show "closed (⋃{i ∈ d. x ∉ i})"
using d' by force
show "x ∉ ⋃{i ∈ d. x ∉ i}"
by auto
qed
then show ?thesis
by force
qed
then obtain k where k: "⋀x. 0 < k x" "⋀i x. ⟦i ∈ d; x ∉ i⟧ ⟹ ball x (k x) ∩ i = {}"
by metis
have "e/2 > 0"
using e by auto
with Henstock_lemma[OF f]
obtain γ where g: "gauge γ"
"⋀p. ⟦p tagged_partial_division_of cbox a b; γ fine p⟧
⟹ (∑(x,k) ∈ p. norm (content k *⇩R f x - integral k f)) < e/2"
by (metis (no_types, lifting))
let ?g = "λx. γ x ∩ 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) ∧ ?g fine p"
then have p: "p tagged_division_of cbox a b" "γ fine p" "(λx. 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. ∃i l. x ∈ i ∧ i ∈ d ∧ (x,l) ∈ p ∧ k = i ∩ l}"
have gp': "γ 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' ⊆ (λ(k, x, l). (x, k ∩ l)) ` (d × p)"
by (force simp: p'_def image_iff)
show "finite ((λ(k, x, l). (x, k ∩ l)) ` (d × p))"
by (simp add: d'(1) p'(1))
qed
next
fix x K
assume "(x, K) ∈ p'"
then have "∃i l. x ∈ i ∧ i ∈ d ∧ (x, l) ∈ p ∧ K = i ∩ l"
unfolding p'_def by auto
then obtain i l where il: "x ∈ i" "i ∈ d" "(x, l) ∈ p" "K = i ∩ l" by blast
show "x ∈ K" and "K ⊆ cbox a b"
using p'(2-3)[OF il(3)] il by auto
show "∃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) ∈ p'"
then have "∃i l. x1 ∈ i ∧ i ∈ d ∧ (x1, l) ∈ p ∧ K1 = i ∩ l"
unfolding p'_def by auto
then obtain i1 l1 where il1: "x1 ∈ i1" "i1 ∈ d" "(x1, l1) ∈ p" "K1 = i1 ∩ l1" by blast
fix x2 K2
assume "(x2,K2) ∈ p'"
then have "∃i l. x2 ∈ i ∧ i ∈ d ∧ (x2, l) ∈ p ∧ K2 = i ∩ l"
unfolding p'_def by auto
then obtain i2 l2 where il2: "x2 ∈ i2" "i2 ∈ d" "(x2, l2) ∈ p" "K2 = i2 ∩ l2" by blast
assume "(x1, K1) ≠ (x2, K2)"
then have "interior i1 ∩ interior i2 = {} ∨ interior l1 ∩ 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 ∩ interior K2 = {}"
unfolding il1 il2 by auto
next
have *: "∀(x, X) ∈ p'. X ⊆ cbox a b"
unfolding p'_def using d' by blast
have "y ∈ ⋃{K. ∃x. (x, K) ∈ p'}" if y: "y ∈ cbox a b" for y
proof -
obtain x l where xl: "(x, l) ∈ p" "y ∈ l"
using y unfolding p'(6)[symmetric] by auto
obtain i where i: "i ∈ d" "y ∈ i"
using y unfolding d'(6)[symmetric] by auto
have "x ∈ 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 ∩ l" in UnionI) (use i xl in auto)
qed
show "⋃{K. ∃x. (x, K) ∈ p'} = cbox a b"
proof
show "⋃{k. ∃x. (x, k) ∈ p'} ⊆ cbox a b"
using * by auto
next
show "cbox a b ⊆ ⋃{k. ∃x. (x, k) ∈ p'}"
proof
fix y
assume y: "y ∈ cbox a b"
obtain x L where xl: "(x, L) ∈ p" "y ∈ L"
using y unfolding p'(6)[symmetric] by auto
obtain I where i: "I ∈ d" "y ∈ I"
using y unfolding d'(6)[symmetric] by auto
have "x ∈ I"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show "y ∈ ⋃{k. ∃x. (x, k) ∈ p'}"
apply (rule_tac X="I ∩ L" in UnionI)
using i xl by (auto simp: p'_def)
qed
qed
qed
then have sum_less_e2: "(∑(x,K) ∈ p'. norm (content K *⇩R f x - integral K f)) < e/2"
using g(2) gp' tagged_division_of_def by blast
have "(x, I ∩ L) ∈ p'" if x: "(x, L) ∈ p" "I ∈ d" and y: "y ∈ I" "y ∈ L"
for x I L y
proof -
have "x ∈ I"
using fineD[OF p(3) that(1)] k(2)[OF ‹I ∈ d›] y by auto
with x have "(∃i l. x ∈ i ∧ i ∈ d ∧ (x, l) ∈ p ∧ I ∩ L = i ∩ l)"
by blast
then have "(x, I ∩ L) ∈ p'"
by (simp add: p'_def)
with y show ?thesis by auto
qed
moreover have "∃y i l. (x, K) = (y, i ∩ l) ∧ (y, l) ∈ p ∧ i ∈ d ∧ i ∩ l ≠ {}"
if xK: "(x,K) ∈ p'" for x K
proof -
obtain i l where il: "x ∈ i" "i ∈ d" "(x, l) ∈ p" "K = i ∩ l"
using xK unfolding p'_def by auto
then show ?thesis
using p'(2) by fastforce
qed
ultimately have p'alt: "p' = {(x, I ∩ L) | x I L. (x,L) ∈ p ∧ I ∈ d ∧ I ∩ L ≠ {}}"
by auto
have sum_p': "(∑(x,K) ∈ p'. norm (integral K f)) = (∑k∈snd ` p'. norm (integral k f))"
apply (subst sum.over_tagged_division_lemma[OF p'',of "λk. 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 × snd ` p)"
by (simp add: d'(1) snd_p(1))
have *: "⋀sni sni' sf sf'. ⟦¦sf' - sni'¦ < e/2; ?S - e/2 < sni; sni' ≤ ?S;
sni ≤ sni'; sf' = sf⟧ ⟹ ¦sf - ?S¦ < e"
by arith
show "norm ((∑(x,k) ∈ p. content k *⇩R norm (f x)) - ?S) < e"
unfolding real_norm_def
proof (rule *)
show "¦(∑(x,K)∈p'. norm (content K *⇩R f x)) - (∑(x,k)∈p'. norm (integral k f))¦ < e/2"
using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
show "(∑(x,k) ∈ p'. norm (integral k f)) ≤?S"
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
show "(∑k∈d. norm (integral k f)) ≤ (∑(x,k) ∈ p'. norm (integral k f))"
proof -
have *: "{k ∩ l | k l. k ∈ d ∧ l ∈ snd ` p} = (λ(k,l). k ∩ l) ` (d × snd ` p)"
by auto
have "(∑K∈d. norm (integral K f)) ≤ (∑i∈d. ∑l∈snd ` p. norm (integral (i ∩ l) f))"
proof (rule sum_mono)
fix K assume k: "K ∈ d"
from d'(4)[OF this] obtain u v where uv: "K = cbox u v" by metis
define d' where "d' = {cbox u v ∩ l |l. l ∈ snd ` p ∧ cbox u v ∩ l ≠ {}}"
have uvab: "cbox u v ⊆ 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 (∑i∈d'. integral i f) ≤ (∑k∈d'. norm (integral k f))"
by (simp add: sum_norm_le)
ultimately have "norm (integral K f) ≤ sum (λk. 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 "… = (∑I∈{K ∩ L |L. L ∈ snd ` p}. norm (integral I f))"
proof -
have *: "norm (integral I f) = 0"
if "I ∈ {cbox u v ∩ l |l. l ∈ snd ` p}"
"I ∉ {cbox u v ∩ l |l. l ∈ snd ` p ∧ cbox u v ∩ l ≠ {}}" 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 "… = (∑l∈snd ` p. norm (integral (K ∩ l) f))"
proof -
have *: "norm (integral (K ∩ l) f) = 0"
if "l ∈ snd ` p" "y ∈ snd ` p" "l ≠ y" "K ∩ l = K ∩ y" for l y
proof -
have "interior (K ∩ l) ⊆ interior (l ∩ y)"
by (metis Int_lower2 interior_mono le_inf_iff that(4))
then have "interior (K ∩ 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) ≤ (∑l∈snd ` p. norm (integral (K ∩ l) f))" .
qed
also have "… = (∑(i,l) ∈ d × snd ` p. norm (integral (i∩l) f))"
by (simp add: sum.cartesian_product)
also have "… = (∑x ∈ d × snd ` p. norm (integral (case_prod op ∩ x) f))"
by (force simp: split_def intro!: sum.cong)
also have "… = (∑k∈{i ∩ l |i l. i ∈ d ∧ l ∈ snd ` p}. norm (integral k f))"
proof -
have eq0: " (integral (l1 ∩ k1) f) = 0"
if "l1 ∩ k1 = l2 ∩ k2" "(l1, k1) ≠ (l2, k2)"
"l1 ∈ d" "(j1,k1) ∈ p" "l2 ∈ d" "(j2,k2) ∈ 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 ‹(j1, k1) ∈ p› ‹l1 ∈ d› d'(4) p'(4) by blast
have "l1 ≠ l2 ∨ k1 ≠ k2"
using that by auto
then have "interior k1 ∩ interior k2 = {} ∨ interior l1 ∩ interior l2 = {}"
by (meson d'(5) old.prod.inject p'(5) that(3) that(4) that(5) that(6))
moreover have "interior (l1 ∩ k1) = interior (l2 ∩ k2)"
by (simp add: that(1))
ultimately have "interior(l1 ∩ 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 "… = (∑(x,k) ∈ p'. norm (integral k f))"
proof -
have 0: "integral (ia ∩ snd (a, b)) f = 0"
if "ia ∩ snd (a, b) ∉ snd ` p'" "ia ∈ d" "(a, b) ∈ p" for ia a b
proof -
have "ia ∩ b = {}"
using that unfolding p'alt image_iff Bex_def not_ex
apply (erule_tac x="(a, ia ∩ b)" in allE)
apply auto
done
then show ?thesis by auto
qed
have 1: "∃i l. snd (a, b) = i ∩ l ∧ i ∈ d ∧ l ∈ snd ` p" if "(a, b) ∈ 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 "(∑(x,k) ∈ p'. norm (content k *⇩R f x)) = (∑(x,k) ∈ p. content k *⇩R norm (f x))"
proof -
let ?S = "{(x, i ∩ l) |x i l. (x, l) ∈ p ∧ i ∈ d}"
have *: "?S = (λ(xl,i). (fst xl, snd xl ∩ i)) ` (p × d)"
by force
have fin_pd: "finite (p × d)"
using finite_cartesian_product[OF p'(1) d'(1)] by metis
have "(∑(x,k) ∈ p'. norm (content k *⇩R f x)) = (∑(x,k) ∈ ?S. ¦content k¦ * 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 "… = (∑((x,l),i)∈p × d. ¦content (l ∩ i)¦ * norm (f x))"
proof -
have "¦content (l1 ∩ k1)¦ * norm (f x1) = 0"
if "(x1, l1) ∈ p" "(x2, l2) ∈ p" "k1 ∈ d" "k2 ∈ d"
"x1 = x2" "l1 ∩ k1 = l2 ∩ k2" "x1 ≠ x2 ∨ l1 ≠ l2 ∨ k1 ≠ 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 ‹(x1, l1) ∈ p› ‹k1 ∈ d› d(1) division_ofD(4) p'(4))
have "l1 ≠ l2 ∨ k1 ≠ k2"
using that by auto
then have "interior k1 ∩ interior k2 = {} ∨ interior l1 ∩ interior l2 = {}"
apply (rule disjE)
using that p'(5) d'(5) by auto
moreover have "interior (l1 ∩ k1) = interior (l2 ∩ k2)"
unfolding that ..
ultimately have "interior (l1 ∩ k1) = {}"
by auto
then show "¦content (l1 ∩ k1)¦ * 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 "… = (∑(x,k) ∈ p. content k *⇩R norm (f x))"
proof -
have sumeq: "(∑i∈d. content (l ∩ i) * norm (f x)) = content l * norm (f x)"
if "(x, l) ∈ 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 "(∑i∈d. ¦content (l ∩ i)¦) = (∑k∈d. content (k ∩ cbox u v))"
by (simp add: Int_commute uv)
also have "… = sum content {k ∩ cbox u v| k. k ∈ d}"
proof -
have eq0: "content (k ∩ cbox u v) = 0"
if "k ∈ d" "y ∈ d" "k ≠ y" and eq: "k ∩ cbox u v = y ∩ cbox u v" for k y
proof -
from d'(4)[OF that(1)] d'(4)[OF that(2)]
obtain α β where α: "k ∩ cbox u v = cbox α β"
by (meson Int_interval)
have "{} = interior ((k ∩ y) ∩ cbox u v)"
by (simp add: d'(5) that)
also have "… = interior (y ∩ (k ∩ cbox u v))"
by auto
also have "… = interior (k ∩ cbox u v)"
unfolding eq by auto
finally show ?thesis
unfolding α content_eq_0_interior ..
qed
then show ?thesis
unfolding Setcompr_eq_image
apply (rule sum.reindex_nontrivial [OF ‹finite d›, unfolded o_def, symmetric])
by auto
qed
also have "… = sum content {cbox u v ∩ k |k. k ∈ d ∧ cbox u v ∩ k ≠ {}}"
apply (rule sum.mono_neutral_right)
unfolding Setcompr_eq_image
apply (rule finite_imageI [OF ‹finite d›])
apply (fastforce simp: inf.commute)+
done
finally show "(∑i∈d. content (l ∩ 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 ⇒ 'm::euclidean_space"
assumes "f integrable_on UNIV"
and "∀d. d division_of (⋃d) ⟶ sum (λk. norm (integral k f)) d ≤ B"
shows "f absolutely_integrable_on UNIV"
proof (rule absolutely_integrable_onI, fact)
let ?f = "λd. ∑k∈d. norm (integral k f)" and ?D = "{d. d division_of (⋃d)}"
have D_1: "?D ≠ {}"
by (rule elementary_interval) auto
have D_2: "bdd_above (?f`?D)"
by (intro bdd_aboveI2[where M=B] assms(2)[rule_format]) simp
note D = D_1 D_2
let ?S = "SUP d:?D. ?f d"
have "⋀a b. f integrable_on cbox a b"
using assms(1) integrable_on_subcbox by blast
then have f_int: "⋀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 "((λx. 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 "∃y∈sum (λk. norm (integral k f)) ` {d. d division_of ⋃d}. ¬ y ≤ ?S - e"
proof (rule ccontr)
assume "¬ ?thesis"
then have "?S ≤ ?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 ⋃d" and "K = (∑k∈d. norm (integral k f))"
"SUPREMUM {d. d division_of ⋃d} (sum (λk. norm (integral k f))) - e < K"
by (auto simp add: image_iff not_le)
then have d: "SUPREMUM {d. d division_of ⋃d} (sum (λk. norm (integral k f))) - e
< (∑k∈d. norm (integral k f))"
by auto
note d'=division_ofD[OF ddiv]
have "bounded (⋃d)"
by (rule elementary_bounded,fact)
from this[unfolded bounded_pos] obtain K where
K: "0 < K" "∀x∈⋃d. norm x ≤ K" by auto
show ?case
proof (intro conjI impI allI exI)
fix a b :: 'n
assume ab: "ball 0 (K + 1) ⊆ cbox a b"
have *: "⋀s s1. ⟦?S - e < s1; s1 ≤ s; s < ?S + e⟧ ⟹ ¦s - ?S¦ < e"
by arith
show "norm (integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0) - ?S) < e"
unfolding real_norm_def
proof (rule * [OF d])
have "(∑k∈d. norm (integral k f)) ≤ sum (λk. integral k (λx. norm (f x))) d"
proof (intro sum_mono)
fix k assume "k ∈ d"
with d'(4) f_int show "norm (integral k f) ≤ integral k (λx. norm (f x))"
by (force simp: absolutely_integrable_on_def integral_norm_bound_integral)
qed
also have "… = integral (⋃d) (λx. 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 "… ≤ integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0)"
proof -
have "⋃d ⊆ 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 "(∑k∈d. norm (integral k f))
≤ integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0)" .
next
have "e/2>0"
using ‹e > 0› by auto
moreover
have f: "f integrable_on cbox a b" "(λx. 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: "⋀p. ⟦p tagged_division_of (cbox a b); d1 fine p⟧ ⟹
norm ((∑(x,k) ∈ p. content k *⇩R norm (f x)) - integral (cbox a b) (λx. norm (f x))) < e/2"
unfolding has_integral_integral has_integral by meson
obtain d2 where "gauge d2"
and d2: "⋀p. ⟦p tagged_partial_division_of (cbox a b); d2 fine p⟧ ⟹
(∑(x,k) ∈ p. norm (content k *⇩R f x - integral k f)) < e/2"
by (blast intro: Henstock_lemma [OF f(1) ‹e/2>0›])
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 ‹gauge d1› ‹gauge d2›], of a b])
(auto simp add: fine_Int)
have *: "⋀sf sf' si di. ⟦sf' = sf; si ≤ ?S; ¦sf - si¦ < e/2;
¦sf' - di¦ < e/2⟧ ⟹ di < ?S + e"
by arith
have "integral (cbox a b) (λx. norm (f x)) < ?S + e"
proof (rule *)
show "¦(∑(x,k)∈p. norm (content k *⇩R f x)) - (∑(x,k)∈p. norm (integral k f))¦ < 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 "¦(∑(x,k) ∈ p. content k *⇩R norm (f x)) - integral (cbox a b) (λx. norm(f x))¦ < e/2"
using d1[OF p(1,2)] by (simp only: real_norm_def)
show "(∑(x,k) ∈ p. content k *⇩R norm (f x)) = (∑(x,k) ∈ p. norm (content k *⇩R f x))"
by (auto simp: split_paired_all sum.cong [OF refl])
show "(∑(x,k) ∈ p. norm (integral k f)) ≤ ?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) (λx. if x ∈ UNIV then norm (f x) else 0) < ?S + e"
by simp
qed
qed (insert K, auto)
qed
then show "(λx. norm (f x)) integrable_on UNIV"
by blast
qed
lemma absolutely_integrable_add[intro]:
fixes f g :: "'n::euclidean_space ⇒ 'm::euclidean_space"
shows "f absolutely_integrable_on s ⟹ g absolutely_integrable_on s ⟹ (λx. f x + g x) absolutely_integrable_on s"
by (rule set_integral_add)
lemma absolutely_integrable_diff[intro]:
fixes f g :: "'n::euclidean_space ⇒ 'm::euclidean_space"
shows "f absolutely_integrable_on s ⟹ g absolutely_integrable_on s ⟹ (λx. f x - g x) absolutely_integrable_on s"
by (rule set_integral_diff)
lemma absolutely_integrable_linear:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
and h :: "'n::euclidean_space ⇒ 'p::euclidean_space"
shows "f absolutely_integrable_on s ⟹ bounded_linear h ⟹ (h ∘ f) absolutely_integrable_on s"
using integrable_bounded_linear[of h lebesgue "λx. indicator s x *⇩R f x"]
by (simp add: linear_simps[of h])
lemma absolutely_integrable_sum:
fixes f :: "'a ⇒ 'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "finite t" and "⋀a. a ∈ t ⟹ (f a) absolutely_integrable_on s"
shows "(λx. sum (λa. f a x) t) absolutely_integrable_on s"
using assms(1,2) by induct auto
lemma absolutely_integrable_integrable_bound:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes le: "∀x∈s. norm (f x) ≤ g x" and f: "f integrable_on s" and g: "g integrable_on s"
shows "f absolutely_integrable_on s"
proof (rule Bochner_Integration.integrable_bound)
show "g absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
proof
show "(λx. norm (g x)) integrable_on s"
using le norm_ge_zero[of "f _"]
by (intro integrable_spike_finite[OF _ _ g, of "{}"])
(auto intro!: abs_of_nonneg intro: order_trans simp del: norm_ge_zero)
qed fact
show "set_borel_measurable lebesgue s f"
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
qed (use le in ‹auto intro!: always_eventually split: split_indicator›)
subsection ‹Componentwise›
proposition absolutely_integrable_componentwise_iff:
shows "f absolutely_integrable_on A ⟷ (∀b∈Basis. (λx. f x ∙ b) absolutely_integrable_on A)"
proof -
have *: "(λx. norm (f x)) integrable_on A ⟷ (∀b∈Basis. (λx. norm (f x ∙ b)) integrable_on A)"
if "f integrable_on A"
proof -
have 1: "⋀i. ⟦(λx. norm (f x)) integrable_on A; i ∈ Basis⟧
⟹ (λx. f x ∙ i) absolutely_integrable_on A"
apply (rule absolutely_integrable_integrable_bound [where g = "λx. norm(f x)"])
using Basis_le_norm integrable_component that apply fastforce+
done
have 2: "∀i∈Basis. (λx. ¦f x ∙ i¦) integrable_on A ⟹ f absolutely_integrable_on A"
apply (rule absolutely_integrable_integrable_bound [where g = "λx. ∑i∈Basis. norm (f x ∙ 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 "(⋀b. b ∈ Basis ⟹ (λx. f x ∙ b) absolutely_integrable_on A) ⟹ f absolutely_integrable_on A"
by (simp add: absolutely_integrable_componentwise_iff)
lemma absolutely_integrable_component:
"f absolutely_integrable_on A ⟹ (λx. f x ∙ (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 ⇒ 'm::euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(λx. c *⇩R f x) absolutely_integrable_on S"
proof -
have "(λx. c *⇩R x) o f absolutely_integrable_on S"
apply (rule absolutely_integrable_linear [OF assms])
by (simp add: bounded_linear_scaleR_right)
then show ?thesis by simp
qed
lemma absolutely_integrable_scaleR_right:
assumes "f absolutely_integrable_on S"
shows "(λx. f x *⇩R c) absolutely_integrable_on S"
using assms by blast
lemma absolutely_integrable_norm:
fixes f :: "'a :: euclidean_space ⇒ 'b :: euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(norm o f) absolutely_integrable_on S"
using assms unfolding absolutely_integrable_on_def by auto
lemma absolutely_integrable_abs:
fixes f :: "'a :: euclidean_space ⇒ 'b :: euclidean_space"
assumes "f absolutely_integrable_on S"
shows "(λx. ∑i∈Basis. ¦f x ∙ i¦ *⇩R i) absolutely_integrable_on S"
(is "?g absolutely_integrable_on S")
proof -
have eq: "?g =
(λx. ∑i∈Basis. ((λy. ∑j∈Basis. if j = i then y *⇩R j else 0) ∘
(λx. norm(∑j∈Basis. if j = i then (x ∙ i) *⇩R j else 0)) ∘ f) x)"
by (simp add: sum.delta)
have *: "(λy. ∑j∈Basis. if j = i then y *⇩R j else 0) ∘
(λx. norm (∑j∈Basis. if j = i then (x ∙ i) *⇩R j else 0)) ∘ f
absolutely_integrable_on S"
if "i ∈ Basis" for i
proof -
have "bounded_linear (λy. ∑j∈Basis. if j = i then y *⇩R j else 0)"
by (simp add: linear_linear algebra_simps linearI)
moreover have "(λx. norm (∑j∈Basis. if j = i then (x ∙ i) *⇩R j else 0)) ∘ f
absolutely_integrable_on S"
unfolding o_def
apply (rule absolutely_integrable_norm [unfolded o_def])
using assms ‹i ∈ Basis›
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 ⇒ real"
assumes f: "f integrable_on A" and "(λx. ¦f x¦) 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: "(λx. ∑i∈Basis. ¦f x ∙ i¦ *⇩R i) integrable_on S"
shows "f absolutely_integrable_on S"
proof -
have "(λx. (f x ∙ i) *⇩R i) absolutely_integrable_on S" if "i ∈ Basis" for i
proof -
have "(λx. ¦f x ∙ i¦) integrable_on S"
using assms integrable_component [OF fcomp, where y=i] that by simp
then have "(λx. f x ∙ i) absolutely_integrable_on S"
apply -
apply (rule abs_absolutely_integrableI_1, auto)
by (simp add: f integrable_component)
then show ?thesis
by (rule absolutely_integrable_scaleR_right)
qed
then have "(λx. ∑i∈Basis. (f x ∙ i) *⇩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 ⟷
f integrable_on S ∧ (λx. ∑i∈Basis. ¦f x ∙ i¦ *⇩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 "(λx. if x ∈ S then ∑i∈Basis. ¦f x ∙ i¦ *⇩R i else 0) = (λx. ∑i∈Basis. ¦(if x ∈ S then f x else 0) ∙ i¦ *⇩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 ⇒ 'm::euclidean_space"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(λx. ∑i∈Basis. max (f x ∙ i) (g x ∙ i) *⇩R i)
absolutely_integrable_on S"
proof -
have "(λx. ∑i∈Basis. max (f x ∙ i) (g x ∙ i) *⇩R i) =
(λx. (1/2) *⇩R (f x + g x + (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i)))"
proof (rule ext)
fix x
have "(∑i∈Basis. max (f x ∙ i) (g x ∙ i) *⇩R i) = (∑i∈Basis. ((f x ∙ i + g x ∙ i + ¦f x ∙ i - g x ∙ i¦) / 2) *⇩R i)"
by (force intro: sum.cong)
also have "... = (1 / 2) *⇩R (∑i∈Basis. (f x ∙ i + g x ∙ i + ¦f x ∙ i - g x ∙ i¦) *⇩R i)"
by (simp add: scaleR_right.sum)
also have "... = (1 / 2) *⇩R (f x + g x + (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i))"
by (simp add: sum.distrib algebra_simps euclidean_representation)
finally
show "(∑i∈Basis. max (f x ∙ i) (g x ∙ i) *⇩R i) =
(1 / 2) *⇩R (f x + g x + (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i))" .
qed
moreover have "(λx. (1 / 2) *⇩R (f x + g x + (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩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 ⇒ real"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(λx. 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 ⇒ 'm::euclidean_space"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(λx. ∑i∈Basis. min (f x ∙ i) (g x ∙ i) *⇩R i)
absolutely_integrable_on S"
proof -
have "(λx. ∑i∈Basis. min (f x ∙ i) (g x ∙ i) *⇩R i) =
(λx. (1/2) *⇩R (f x + g x - (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i)))"
proof (rule ext)
fix x
have "(∑i∈Basis. min (f x ∙ i) (g x ∙ i) *⇩R i) = (∑i∈Basis. ((f x ∙ i + g x ∙ i - ¦f x ∙ i - g x ∙ i¦) / 2) *⇩R i)"
by (force intro: sum.cong)
also have "... = (1 / 2) *⇩R (∑i∈Basis. (f x ∙ i + g x ∙ i - ¦f x ∙ i - g x ∙ i¦) *⇩R i)"
by (simp add: scaleR_right.sum)
also have "... = (1 / 2) *⇩R (f x + g x - (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i))"
by (simp add: sum.distrib sum_subtractf algebra_simps euclidean_representation)
finally
show "(∑i∈Basis. min (f x ∙ i) (g x ∙ i) *⇩R i) =
(1 / 2) *⇩R (f x + g x - (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩R i))" .
qed
moreover have "(λx. (1 / 2) *⇩R (f x + g x - (∑i∈Basis. ¦f x ∙ i - g x ∙ i¦ *⇩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 ⇒ real"
assumes "f absolutely_integrable_on S" "g absolutely_integrable_on S"
shows "(λx. 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 ⇒ 'b :: euclidean_space"
assumes "f integrable_on A" and comp: "⋀x b. ⟦x ∈ A; b ∈ Basis⟧ ⟹ 0 ≤ f x ∙ b"
shows "f absolutely_integrable_on A"
proof -
have "(λx. (f x ∙ i) *⇩R i) absolutely_integrable_on A" if "i ∈ Basis" for i
proof -
have "(λx. f x ∙ i) integrable_on A"
by (simp add: assms(1) integrable_component)
then have "(λx. f x ∙ 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 "(λx. ∑i∈Basis. (f x ∙ i) *⇩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 ⇒ 'b :: euclidean_space"
assumes f: "f integrable_on A" and g: "g absolutely_integrable_on A"
and comp: "⋀x b. ⟦x ∈ A; b ∈ Basis⟧ ⟹ f x ∙ b ≤ g x ∙ b"
shows "f absolutely_integrable_on A"
proof -
have "(λx. 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 ⇒ 'b :: euclidean_space"
assumes f: "f absolutely_integrable_on A" and g: "g integrable_on A"
and comp: "⋀x b. ⟦x ∈ A; b ∈ Basis⟧ ⟹ f x ∙ b ≤ g x ∙ b"
shows "g absolutely_integrable_on A"
proof -
have "(λx. f x + (g x - f x)) absolutely_integrable_on A"
apply (rule absolutely_integrable_add [OF f nonnegative_absolutely_integrable])
using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
by (simp add: comp inner_diff_left)
then show ?thesis
by simp
qed
subsection ‹Dominated convergence›
lemma dominated_convergence:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ 'm::euclidean_space"
assumes f: "⋀k. (f k) integrable_on s" and h: "h integrable_on s"
and le: "⋀k. ∀x ∈ s. norm (f k x) ≤ h x"
and conv: "∀x ∈ s. (λk. f k x) ⇢ g x"
shows "g integrable_on s" "(λk. integral s (f k)) ⇢ integral s g"
proof -
have 3: "h absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
proof
show "(λx. norm (h x)) integrable_on s"
proof (intro integrable_spike_finite[OF _ _ h, of "{}"] ballI)
fix x assume "x ∈ s - {}" then show "norm (h x) = h x"
by (metis Diff_empty abs_of_nonneg bot_set_def le norm_ge_zero order_trans real_norm_def)
qed auto
qed fact
have 2: "set_borel_measurable lebesgue s (f k)" for k
using f by (auto intro: has_integral_implies_lebesgue_measurable simp: integrable_on_def)
then have 1: "set_borel_measurable lebesgue s g"
by (rule borel_measurable_LIMSEQ_metric) (use conv in ‹auto split: split_indicator›)
have 4: "AE x in lebesgue. (λi. indicator s x *⇩R f i x) ⇢ indicator s x *⇩R g x"
"AE x in lebesgue. norm (indicator s x *⇩R f k x) ≤ indicator s x *⇩R h x" for k
using conv le by (auto intro!: always_eventually split: split_indicator)
have g: "g absolutely_integrable_on s"
using 1 2 3 4 by (rule integrable_dominated_convergence)
then show "g integrable_on s"
by (auto simp: absolutely_integrable_on_def)
have "(λk. (LINT x:s|lebesgue. f k x)) ⇢ (LINT x:s|lebesgue. g x)"
using 1 2 3 4 by (rule integral_dominated_convergence)
then show "(λk. integral s (f k)) ⇢ 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 ⇒ 'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "⋀k. (f k has_integral y k) s" "h integrable_on s"
"⋀k. ∀x∈s. norm (f k x) ≤ h x" "∀x∈s. (λk. f k x) ⇢ g x"
and x: "y ⇢ x"
shows "(g has_integral x) s"
proof -
have int_f: "⋀k. (f k) integrable_on s"
using assms by (auto simp: integrable_on_def)
have "(g has_integral (integral s g)) s"
by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
moreover have "integral s g = x"
proof (rule LIMSEQ_unique)
show "(λi. integral s (f i)) ⇢ x"
using integral_unique[OF assms(1)] x by simp
show "(λi. integral s (f i)) ⇢ integral s g"
by (intro dominated_convergence[OF int_f assms(2)]) fact+
qed
ultimately show ?thesis
by simp
qed
subsection ‹Fundamental Theorem of Calculus for the Lebesgue integral›
text ‹
For the positive integral we replace continuity with Borel-measurability.
›
lemma
fixes f :: "real ⇒ real"
assumes [measurable]: "f ∈ borel_measurable borel"
assumes f: "⋀x. x ∈ {a..b} ⟹ DERIV F x :> f x" "⋀x. x ∈ {a..b} ⟹ 0 ≤ f x" and "a ≤ b"
shows nn_integral_FTC_Icc: "(∫⇧+x. ennreal (f x) * indicator {a .. b} x ∂lborel) = F b - F a" (is ?nn)
and has_bochner_integral_FTC_Icc_nonneg:
"has_bochner_integral lborel (λx. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
and integral_FTC_Icc_nonneg: "(∫x. f x * indicator {a .. b} x ∂lborel) = F b - F a" (is ?eq)
and integrable_FTC_Icc_nonneg: "integrable lborel (λx. f x * indicator {a .. b} x)" (is ?int)
proof -
have *: "(λx. f x * indicator {a..b} x) ∈ borel_measurable borel" "⋀x. 0 ≤ f x * indicator {a..b} x"
using f(2) by (auto split: split_indicator)
have F_mono: "a ≤ x ⟹ x ≤ y ⟹ y ≤ b⟹ F x ≤ 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)] ‹a ≤ b›)
then have i: "((λx. f x * indicator {a .. b} x) has_integral F b - F a) UNIV"
unfolding indicator_def if_distrib[where f="λx. a * x" for a]
by (simp cong del: if_weak_cong del: atLeastAtMost_iff)
then have nn: "(∫⇧+x. f x * indicator {a .. b} x ∂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 ‹a ≤ b›)
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 ⇒ 'a :: euclidean_space"
assumes "a ≤ b"
assumes "⋀x. a ≤ x ⟹ x ≤ b ⟹ (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 (λx. indicator {a .. b} x *⇩R f x) (F b - F a)" (is ?has)
and integral_FTC_Icc: "(∫x. indicator {a .. b} x *⇩R f x ∂lborel) = F b - F a" (is ?eq)
proof -
let ?f = "λx. indicator {a .. b} x *⇩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⇧L lborel ?f) {a..b}"
using has_integral_integral_lborel[OF int]
unfolding indicator_def if_distrib[where f="λx. x *⇩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 ⇒ real"
assumes "a ≤ b"
assumes deriv: "⋀x. a ≤ x ⟹ x ≤ b ⟹ DERIV F x :> f x"
assumes cont: "⋀x. a ≤ x ⟹ x ≤ b ⟹ isCont f x"
shows has_bochner_integral_FTC_Icc_real:
"has_bochner_integral lborel (λx. f x * indicator {a .. b} x) (F b - F a)" (is ?has)
and integral_FTC_Icc_real: "(∫x. f x * indicator {a .. b} x ∂lborel) = F b - F a" (is ?eq)
proof -
have 1: "⋀x. a ≤ x ⟹ x ≤ b ⟹ (F has_vector_derivative f x) (at x within {a .. b})"
unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
using deriv by (auto intro: DERIV_subset)
have 2: "continuous_on {a .. b} f"
using cont by (intro continuous_at_imp_continuous_on) auto
show ?has ?eq
using has_bochner_integral_FTC_Icc[OF ‹a ≤ b› 1 2] integral_FTC_Icc[OF ‹a ≤ b› 1 2]
by (auto simp: mult.commute)
qed
lemma nn_integral_FTC_atLeast:
fixes f :: "real ⇒ real"
assumes f_borel: "f ∈ borel_measurable borel"
assumes f: "⋀x. a ≤ x ⟹ DERIV F x :> f x"
assumes nonneg: "⋀x. a ≤ x ⟹ 0 ≤ f x"
assumes lim: "(F ⤏ T) at_top"
shows "(∫⇧+x. ennreal (f x) * indicator {a ..} x ∂lborel) = T - F a"
proof -
let ?f = "λ(i::nat) (x::real). ennreal (f x) * indicator {a..a + real i} x"
let ?fR = "λx. ennreal (f x) * indicator {a ..} x"
have F_mono: "a ≤ x ⟹ x ≤ y ⟹ F x ≤ 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 ≤ x ⟹ F x ≤ T" for x
by (intro tendsto_lowerbound[OF lim])
(auto simp: eventually_at_top_linorder)
have "(SUP i::nat. ?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 (λn. ?f n x = ?fR x) sequentially"
by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
then show "(λn. ?f n x) ⇢ ?fR x"
by (rule Lim_eventually)
qed (auto simp: nonneg incseq_def le_fun_def split: split_indicator)
then have "integral⇧N lborel ?fR = (∫⇧+ x. (SUP i::nat. ?f i x) ∂lborel)"
by simp
also have "… = (SUP i::nat. (∫⇧+ x. ?f i x ∂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 "⋀i. (?f i) ∈ borel_measurable lborel"
using f_borel by auto
qed
also have "… = (SUP i::nat. ennreal (F (a + real i) - F a))"
by (subst nn_integral_FTC_Icc[OF f_borel f nonneg]) auto
also have "… = T - F a"
proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
have "(λx. F (a + real x)) ⇢ T"
apply (rule filterlim_compose[OF lim filterlim_tendsto_add_at_top])
apply (rule LIMSEQ_const_iff[THEN iffD2, OF refl])
apply (rule filterlim_real_sequentially)
done
then show "(λn. ennreal (F (a + real n) - F a)) ⇢ 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 ≤ b ⟹ (∫x. x^k * indicator {a..b} x ∂lborel) = (b^Suc k - a^Suc k) / Suc k"
proof (subst integral_FTC_Icc_real)
fix x show "DERIV (λx. 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 ‹Integration by parts›
lemma integral_by_parts_integrable:
fixes f g F G::"real ⇒ real"
assumes "a ≤ b"
assumes cont_f[intro]: "!!x. a ≤x ⟹ x≤b ⟹ isCont f x"
assumes cont_g[intro]: "!!x. a ≤x ⟹ x≤b ⟹ isCont g x"
assumes [intro]: "!!x. DERIV F x :> f x"
assumes [intro]: "!!x. DERIV G x :> g x"
shows "integrable lborel (λx.((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 ⇒ real"
assumes [arith]: "a ≤ b"
assumes cont_f[intro]: "!!x. a ≤x ⟹ x≤b ⟹ isCont f x"
assumes cont_g[intro]: "!!x. a ≤x ⟹ x≤b ⟹ isCont g x"
assumes [intro]: "!!x. DERIV F x :> f x"
assumes [intro]: "!!x. DERIV G x :> g x"
shows "(∫x. (F x * g x) * indicator {a .. b} x ∂lborel)
= F b * G b - F a * G a - ∫x. (f x * G x) * indicator {a .. b} x ∂lborel"
proof-
have 0: "(∫x. (F x * g x + f x * G x) * indicator {a .. b} x ∂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 "(∫x. (F x * g x + f x * G x) * indicator {a .. b} x ∂lborel) =
(∫x. (F x * g x) * indicator {a .. b} x ∂lborel) + ∫x. (f x * G x) * indicator {a .. b} x ∂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 ⇒ real"
assumes "a ≤ b"
assumes "!!x. a ≤x ⟹ x≤b ⟹ isCont f x"
assumes "!!x. a ≤x ⟹ x≤b ⟹ isCont g x"
assumes "!!x. DERIV F x :> f x"
assumes "!!x. DERIV G x :> g x"
shows "(∫x. indicator {a .. b} x *⇩R (F x * g x) ∂lborel)
= F b * G b - F a * G a - ∫x. indicator {a .. b} x *⇩R (f x * G x) ∂lborel"
using integral_by_parts[OF assms] by (simp add: ac_simps)
lemma has_bochner_integral_even_function:
fixes f :: "real ⇒ 'a :: {banach, second_countable_topology}"
assumes f: "has_bochner_integral lborel (λx. indicator {0..} x *⇩R f x) x"
assumes even: "⋀x. f (- x) = f x"
shows "has_bochner_integral lborel f (2 *⇩R x)"
proof -
have indicator: "⋀x::real. indicator {..0} (- x) = indicator {0..} x"
by (auto split: split_indicator)
have "has_bochner_integral lborel (λx. indicator {.. 0} x *⇩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 (λx. indicator {0..} x *⇩R f x + indicator {.. 0} x *⇩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 ⇒ 'a :: {banach, second_countable_topology}"
assumes f: "has_bochner_integral lborel (λx. indicator {0..} x *⇩R f x) x"
assumes odd: "⋀x. f (- x) = - f x"
shows "has_bochner_integral lborel f 0"
proof -
have indicator: "⋀x::real. indicator {..0} (- x) = indicator {0..} x"
by (auto split: split_indicator)
have "has_bochner_integral lborel (λx. - indicator {.. 0} x *⇩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 (λx. indicator {.. 0} x *⇩R f x) (- x)"
by simp
with f have "has_bochner_integral lborel (λx. indicator {0..} x *⇩R f x + indicator {.. 0} x *⇩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 ⇒ real"
assumes f: "continuous_on (closure S) f"
and nonneg_interior: "⋀x. x ∈ S ⟹ 0 ≤ f x"
and pos: "0 < emeasure lborel S"
and finite: "emeasure lborel S < ∞"
and regular: "emeasure lborel (closure S) = emeasure lborel S"
and opn: "open S"
assumes int: "(f has_integral 0) (closure S)"
assumes x: "x ∈ 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 S› )
have nonneg: "0 ≤ f x" if "x ∈ 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 "… = 0 ⟷ (AE x in lebesgue. indicator (closure S) x *⇩R f x = 0)"
by (rule integral_nonneg_eq_0_iff_AE[OF af]) (use nonneg in ‹auto simp: indicator_def›)
also have "… ⟷ (AE x in lebesgue. x ∈ {x. x ∈ closure S ⟶ f x = 0})"
by (auto simp: indicator_def)
finally have "(AE x in lebesgue. x ∈ {x. x ∈ closure S ⟶ f x = 0})" by simp
moreover have "(AE x in lebesgue. x ∈ - frontier S)"
using zero
by (auto simp: eventually_ae_filter null_sets_def intro!: exI[where x="frontier S"])
ultimately have ae: "AE x ∈ S in lebesgue. x ∈ {x ∈ closure S. f x = 0}" (is ?th)
by eventually_elim (use closure_subset in ‹auto simp: ›)
have "closed {0::real}" by simp
with continuous_on_closed_vimage[OF closed_closure, of S f] f
have "closed (f -` {0} ∩ closure S)" by blast
then have "closed {x ∈ closure S. f x = 0}" by (auto simp: vimage_def Int_def conj_commute)
with ‹open S› have "x ∈ {x ∈ closure S. f x = 0}" if "x ∈ S" for x using ae that
by (rule mem_closed_if_AE_lebesgue_open)
then have "f x = 0" if "x ∈ S" for x using that by auto
from continuous_constant_on_closure[OF f this ‹x ∈ closure S›]
show "f x = 0" .
qed
lemma has_integral_0_cbox_imp_0:
fixes f :: "'a::euclidean_space ⇒ real"
assumes f: "continuous_on (cbox a b) f"
and nonneg_interior: "⋀x. x ∈ box a b ⟹ 0 ≤ f x"
assumes int: "(f has_integral 0) (cbox a b)"
assumes ne: "box a b ≠ {}"
assumes x: "x ∈ cbox a b"
shows "f x = 0"
proof -
have "0 < emeasure lborel (box a b)"
using ne x unfolding emeasure_lborel_box_eq
by (force intro!: prod_pos simp: mem_box algebra_simps)
then show ?thesis using assms
by (intro has_integral_0_closure_imp_0[of "box a b" f x])
(auto simp: emeasure_lborel_box_eq emeasure_lborel_cbox_eq algebra_simps mem_box)
qed
end