# Theory Equivalence_Lebesgue_Henstock_Integration

```(*  Title:      HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author:     Johannes Hölzl, TU München
Author:     Robert Himmelmann, TU München
Huge cleanup by LCP
*)

theory Equivalence_Lebesgue_Henstock_Integration
imports
Lebesgue_Measure
Henstock_Kurzweil_Integration
Complete_Measure
Set_Integral
Homeomorphism
Cartesian_Euclidean_Space
begin

lemma LIMSEQ_if_less: "(λk. if i < k then a else b) ⇢ a"
by (rule_tac k="Suc i" in LIMSEQ_offset) auto

text ‹Note that the rhs is an implication. This lemma plays a specific role in one proof.›
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"
using assms by metric

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: field_split_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
by (cases "?μ (E ∩ ⋃{k∈snd`p. k ∩ C X m = {}})") (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: top_unique simp flip: 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 ∈ ⋃(snd ` (p - s))"
by (auto intro!: bexI[of _ "(z, k)"] elim: ballE[of _ _ "(z, k)"])
qed (auto intro!: sets.Int sets.finite_UN ennreal_mono_minus intro: p_in_L)
also have "… = 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))"
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)) ⊆ ⋃(B ` UNIV)"
unfolding B_def by (intro UN_mono box_subset_cbox order_refl)
then show "countable (range B)" "space lebesgue ⊆ ⋃(B ` UNIV)"
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)"
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
using has_integral_const[of "1::real" l u]
by (simp flip: has_integral_restrict[OF box_subset_cbox] add: has_integral_spike_interior)
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"

have "((λx. 1) has_integral ?M (box a b)) (box a b)"
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"
have "((λx. if x ∈ ⋃(F ` UNIV) ∩ 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"
show "(λk. ?f k x) ⇢ (if x ∈ ⋃(F ` UNIV) ∩ box a b then 1 else 0)" for x
by (auto simp: * sum.If_cases Iio_Int_singleton if_distrib LIMSEQ_if_less cong: if_cong)
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 of_bool_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
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
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
then have "(∫⇧+ x. h x + g x ∂lborel) = (∫⇧+ x. h x ∂lborel) + (∫⇧+ x. g x ∂lborel)"
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)
next
case (seq U)
note seq(1)[measurable] and f[measurable]

have U_le_f: "U i x ≤ f x" for i x
by (metis (no_types) LIMSEQ_le_const UNIV_I incseq_def le_fun_def seq.hyps(4) seq.hyps(5) space_borel)

{ 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
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
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 of_bool_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
qed
with emeasure_A show ?thesis
by auto
next
case False
then have "((λx. 1) has_integral measure lborel A) A"
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
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"
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. x ∈ Ω ⟹ 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 of_bool_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
also have "(λx. abs (indicator Ω x * f x)) = (λx. indicator Ω x * f x)"
using nonneg by (auto simp: indicator_def fun_eq_iff)
finally show ?thesis .
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

lemma set_nn_integral_lborel_eq_integral:
fixes f::"'a::euclidean_space ⇒ real"
assumes "set_borel_measurable borel A f"
assumes "⋀x. x ∈ A ⟹ 0 ≤ f x" "(∫⇧+x∈A. f x ∂lborel) < ∞"
shows "(∫⇧+x∈A. f x ∂lborel) = integral A f"
proof -
have eq: "(∫⇧+x∈A. f x ∂lborel) = (∫⇧+x. ennreal (indicator A x * f x) ∂lborel)"
by (intro nn_integral_cong) (auto simp: indicator_def)
also have "… = integral UNIV (λx. indicator A x * f x)"
using assms eq by (intro nn_integral_lborel_eq_integral)
(auto simp: indicator_def set_borel_measurable_def)
also have "integral UNIV (λx. indicator A x * f x) = integral A (λx. indicator A x * f x)"
by (rule integral_spike_set) (auto intro: empty_imp_negligible)

also have "… = integral A f"
by (rule integral_cong) (auto simp: indicator_def)
finally show ?thesis .
qed

lemma nn_integral_has_integral_lebesgue':
fixes f :: "'a::euclidean_space ⇒ real"
assumes nonneg: "⋀x. x ∈ Ω ⟹ 0 ≤ f x" and I: "(f has_integral I) Ω"
shows "integral⇧N lborel (λx. ennreal (f x) * indicator Ω x) = ennreal I"
proof -
have "integral⇧N lborel (λx. ennreal (f x) * indicator Ω x) =
integral⇧N lborel (λx. ennreal (indicator Ω x * f x))"
by (intro nn_integral_cong) (auto simp: indicator_def)
also have "… = ennreal I"
using assms by (intro nn_integral_has_integral_lebesgue)
finally show ?thesis .
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"
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"
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 has_integral_integral_lebesgue_on:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "integrable (lebesgue_on S) f" "S ∈ sets lebesgue"
shows "(f has_integral (integral⇧L (lebesgue_on S) f)) S"
proof -
let ?f = "λx. if x ∈ S then f x else 0"
have "integrable lebesgue (λx. indicat_real S x *⇩R f x)"
using indicator_scaleR_eq_if [of S _ f] assms
by (metis (full_types) integrable_restrict_space sets.Int_space_eq2)
then have "integrable lebesgue ?f"
using indicator_scaleR_eq_if [of S _ f] assms by auto
then have "(?f has_integral (integral⇧L lebesgue ?f)) UNIV"
by (rule has_integral_integral_lebesgue)
then have "(f has_integral (integral⇧L lebesgue ?f)) S"
using has_integral_restrict_UNIV by blast
moreover
have "S ∩ space lebesgue ∈ sets lebesgue"
then have "(integral⇧L lebesgue ?f) = (integral⇧L (lebesgue_on S) f)"
ultimately show ?thesis
by auto
qed

lemma lebesgue_integral_eq_integral:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "integrable (lebesgue_on S) f" "S ∈ sets lebesgue"
shows "integral⇧L (lebesgue_on S) f = integral S f"
by (metis has_integral_integral_lebesgue_on assms integral_unique)

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))"
unfolding set_lebesgue_integral_def
by (subst lborel_integral_real_affine[where c="-1" and t=0])
(auto intro!: Bochner_Integration.integral_cong split: split_indicator)

lemma borel_integrable_atLeastAtMost':
fixes f :: "real ⇒ 'a::{banach, second_countable_topology}"
assumes f: "continuous_on {a..b} f"
shows "set_integrable lborel {a..b} f"
unfolding set_integrable_def
by (intro borel_integrable_compact compact_Icc f)

lemma integral_FTC_atLeastAtMost:
fixes f :: "real ⇒ '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]
unfolding set_integrable_def by (rule has_integral_integral_lborel)
moreover
have "(f has_integral F b - F a) {a .. b}"
by (intro fundamental_theorem_of_calculus ballI assms) auto
then have "(?f has_integral F b - F a) {a .. b}"
by (subst has_integral_cong[where g=f]) auto
then have "(?f has_integral F b - F a) UNIV"
by (intro has_integral_on_superset[where T=UNIV and S="{a..b}"]) auto
ultimately show "integral⇧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"
using assms has_integral_integral_lborel
unfolding set_integrable_def set_lebesgue_integral_def by blast
hence 1: "(f has_integral (set_lebesgue_integral lborel S f)) S"
thus "f integrable_on S"
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 f
by (fastforce simp add: set_integrable_def set_lebesgue_integral_def indicator_def
of_bool_def if_distrib[of "λx. x *⇩R f _"] 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_zero [simp]: "(λx. 0) absolutely_integrable_on S"

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))"
using integrable_norm set_integrable_def by blast
show "f integrable_on S"
by (rule set_lebesgue_integral_eq_integral[OF f])
have "(λx. norm (indicator S x *⇩R f x)) = (λx. if x ∈ S then norm (f x) else 0)"
by auto
with integrable_on_lebesgue[OF nf] show "(λx. norm (f x)) integrable_on S"
next
assume f: "f integrable_on S" and nf: "(λx. norm (f x)) integrable_on S"
show "f absolutely_integrable_on S"
unfolding set_integrable_def
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)"]
by (auto simp: integrable_on_def nn_integral_completion)
qed
qed

lemma integrable_on_lebesgue_on:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "integrable (lebesgue_on S) f" and S: "S ∈ sets lebesgue"
shows "f integrable_on S"
proof -
have "integrable lebesgue (λx. indicator S x *⇩R f x)"
using S f inf_top.comm_neutral integrable_restrict_space by blast
then show ?thesis
using absolutely_integrable_on_def set_integrable_def by blast
qed

lemma absolutely_integrable_imp_integrable:
assumes "f absolutely_integrable_on S" "S ∈ sets lebesgue"
shows "integrable (lebesgue_on S) f"
by (meson assms integrable_restrict_space set_integrable_def sets.Int sets.top)

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"
unfolding set_integrable_def
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"
by (rule absolutely_integrable_onI [OF f]) (use assms in ‹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"
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 absolutely_integrable_on_scaleR_iff:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows
"(λx. c *⇩R f x) absolutely_integrable_on S ⟷
c = 0 ∨ f absolutely_integrable_on S"
proof (cases "c=0")
case False
then show ?thesis
unfolding absolutely_integrable_on_def
qed auto

lemma absolutely_integrable_spike:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "f absolutely_integrable_on T" and S: "negligible S" "⋀x. x ∈ T - S ⟹ g x = f x"
shows "g absolutely_integrable_on T"
using assms integrable_spike
unfolding absolutely_integrable_on_def by metis

lemma absolutely_integrable_negligible:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "negligible S"
shows "f absolutely_integrable_on S"
using assms by (simp add: absolutely_integrable_on_def integrable_negligible)

lemma absolutely_integrable_spike_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "negligible S" "⋀x. x ∈ T - S ⟹ g x = f x"
shows "(f absolutely_integrable_on T ⟷ g absolutely_integrable_on T)"
using assms by (blast intro: absolutely_integrable_spike sym)

lemma absolutely_integrable_spike_set_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "negligible {x ∈ S - T. f x ≠ 0}" "negligible {x ∈ T - S. f x ≠ 0}"
shows "(f absolutely_integrable_on S ⟷ f absolutely_integrable_on T)"
proof -
have "(λx. if x ∈ S then f x else 0) absolutely_integrable_on UNIV ⟷
(λx. if x ∈ T then f x else 0) absolutely_integrable_on UNIV"
proof (rule absolutely_integrable_spike_eq)
show "negligible ({x ∈ S - T. f x ≠ 0} ∪ {x ∈ T - S. f x ≠ 0})"
by (rule negligible_Un [OF assms])
qed auto
with absolutely_integrable_restrict_UNIV show ?thesis
by blast
qed

lemma absolutely_integrable_spike_set:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes f: "f absolutely_integrable_on S" and neg: "negligible {x ∈ S - T. f x ≠ 0}" "negligible {x ∈ T - S. f x ≠ 0}"
shows "f absolutely_integrable_on T"
using absolutely_integrable_spike_set_eq f neg by blast

lemma absolutely_integrable_reflect[simp]:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "(λx. f(-x)) absolutely_integrable_on cbox (-b) (-a) ⟷ f absolutely_integrable_on cbox a b"
unfolding absolutely_integrable_on_def
by (metis (mono_tags, lifting) integrable_eq integrable_reflect)

lemma absolutely_integrable_reflect_real[simp]:
fixes f :: "real ⇒ 'b::euclidean_space"
shows "(λx. f(-x)) absolutely_integrable_on {-b .. -a} ⟷ f absolutely_integrable_on {a..b::real}"
unfolding box_real[symmetric] by (rule absolutely_integrable_reflect)

lemma absolutely_integrable_on_subcbox:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
shows "⟦f absolutely_integrable_on S; cbox a b ⊆ S⟧ ⟹ f absolutely_integrable_on cbox a b"
by (meson absolutely_integrable_on_def integrable_on_subcbox)

lemma absolutely_integrable_on_subinterval:
fixes f :: "real ⇒ 'b::euclidean_space"
shows "⟦f absolutely_integrable_on S; {a..b} ⊆ S⟧ ⟹ f absolutely_integrable_on {a..b}"
using absolutely_integrable_on_subcbox by fastforce

lemma integrable_subinterval:
fixes f :: "real ⇒ 'a::euclidean_space"
assumes "integrable (lebesgue_on {a..b}) f"
and "{c..d} ⊆ {a..b}"
shows "integrable (lebesgue_on {c..d}) f"
proof (rule absolutely_integrable_imp_integrable)
show "f absolutely_integrable_on {c..d}"
proof -
have "f integrable_on {c..d}"
using assms integrable_on_lebesgue_on integrable_on_subinterval by fastforce
moreover have "(λx. norm (f x)) integrable_on {c..d}"
proof (rule integrable_on_subinterval)
show "(λx. norm (f x)) integrable_on {a..b}"
qed (use assms in auto)
ultimately show ?thesis
by (auto simp: absolutely_integrable_on_def)
qed
qed auto

lemma indefinite_integral_continuous_real:
fixes f :: "real ⇒ 'a::euclidean_space"
assumes "integrable (lebesgue_on {a..b}) f"
shows "continuous_on {a..b} (λx. integral⇧L (lebesgue_on {a..x}) f)"
proof -
have "f integrable_on {a..b}"
then have "continuous_on {a..b} (λx. integral {a..x} f)"
using indefinite_integral_continuous_1 by blast
moreover have "integral⇧L (lebesgue_on {a..x}) f = integral {a..x} f" if "a ≤ x" "x ≤ b" for x
proof -
have "{a..x} ⊆ {a..b}"
using that by auto
then have "integrable (lebesgue_on {a..x}) f"
using integrable_subinterval assms by blast
then show "integral⇧L (lebesgue_on {a..x}) f = integral {a..x} f"
qed
ultimately show ?thesis
by (metis (no_types, lifting) atLeastAtMost_iff continuous_on_cong)
qed

lemma lmeasurable_iff_integrable_on: "S ∈ lmeasurable ⟷ (λx. 1::real) integrable_on S"
by (subst absolutely_integrable_on_iff_nonneg[symmetric])

lemma lmeasure_integral_UNIV: "S ∈ lmeasurable ⟹ measure lebesgue S = integral UNIV (indicator S)"