Báo cáo hóa học: "MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES" ppt

GUSEINOVReceived 26 January 2006; Revised 17 April 2006; Accepted 18 April 2006 We study the process of multiple Lebesgue integration on time scales.. In [3], we presented the process of

MARTIN BOHNER AND GUSEIN SH GUSEINOV

Received 26 January 2006; Revised 17 April 2006; Accepted 18 April 2006

We study the process of multiple Lebesgue integration on time scales The relationship of the Riemann and the Lebesgue multiple integrals is investigated

Copyright © 2006 M Bohner and G Sh Guseinov This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations For single variable differential and integral calculus on time scales, we refer the reader to the textbooks [4,5] and the references given therein Multivariable calcu-lus on time scales was developed by the authors [2,3] In [3], we presented the process

of Riemann multiple delta (nabla and mixed types) integration on time scales In the present paper, we introduce the definitions of Lebesgue multi-dimensional delta (nabla and mixed types) measures and integrals on time scales A comparison of the Lebesgue multiple delta integral with the Riemann multiple delta integral is given

Beside this introductory section, this paper consists of two sections InSection 2, fol-lowing [3], we give the Darboux definition of the Riemann multiple delta integral and present some needed facts connected to it The main part of this paper isSection 3 There,

a brief description of the Carath´eodory construction of a Lebesgue measure in an ab-stract setting is given Then the Lebesgue multi-dimensional delta measure on time scales

is introduced and the Lebesgue delta measure of any single-point set is calculated When

we have a measure, integration theory is available according to the well-known general scheme of the Lebesgue integration process Finally, we compare the Lebesgue multiple delta integral with the Riemann multiple delta integral We indicate also a way to define, along with the Lebesgue multi-dimensional delta measure, the nabla and mixed types Lebesgue multi-dimensional measures on time scales

Hindawi Publishing Corporation

Advances in Di fference Equations

Volume 2006, Article ID 26391, Pages 1 12

DOI 10.1155/ADE/2006/26391

2 Multiple Riemann integration

In this section, following [3], we give the definition of the multiple Riemann integral

on time scales over arbitrary bounded regions as we will compare in the next section the Riemann integral with the Lebesgue integral introduced therein For convenience, we present the exposition for functions of two independent variables

LetT 1 andT 2 be two time scales Fori =1, 2, letσ i,ρ i, andΔi denote the forward jump operator, the backward jump operator, and the delta differentiation operator, re-spectively, onTi Supposea < b are points inT 1,c < d are points inT 2, [a,b) is the

half-closed bounded interval inT 1, and [c,d) is the half-closed bounded interval inT 2 Let us introduce a “rectangle” (or “delta rectangle”) inT 1× T2by

R =[a,b) ×[c,d) =(t,s) : t ∈[a,b), s ∈[c,d)

First we define Riemann integrals over rectangles of the type given in (2.1) Let



t0,t1, ,t n

⊂[a,b], wherea = t0< t1< ··· < t n = b,



s0,s1, ,s k



⊂[c,d], wherec = s0< s1< ··· < s k = d. (2.2)

The numbersn and k may be arbitrary positive integers We call the collection of intervals

P1=t i −1,t i

: 1≤ i ≤ n

(2.3)

aΔ-partition (or delta partition) of [a,b) and denote the set of all Δ-partitions of [a,b) by ᏼ([a,b)) Similarly, the collection of intervals

P2=s j −1,s j

: 1≤ j ≤ k

(2.4)

is called a Δ-partition of [c,d) and the set of all Δ-partitions of [c,d) is denoted by ᏼ([c,d)) Let us set

R i j =t i −1,t i



×s j −1,s j



, where 1≤ i ≤ n, 1 ≤ j ≤ k. (2.5)

We call the collection

P =R i j: 1≤ i ≤ n,1 ≤ j ≤ k

(2.6)

aΔ-partition of R, generated by the Δ-partitions P1andP2of [a,b) and [c,d), respectively,

and writeP = P1× P2 The rectanglesR i j, 1≤ i ≤ n, 1 ≤ j ≤ k, are called the subrectangles

of the partitionP The set of all Δ-partitions of R is denoted by ᏼ(R).

Let f : R → Rbe a bounded function We set

M =sup

f (t,s) : (t,s) ∈ R

, m =inf

f (t,s) : (t,s) ∈ R

(2.7) and for 1≤ i ≤ n, 1 ≤ j ≤ k,

M i j =sup

f (t,s) : (t,s) ∈ R i j

, m i j =inf

f (t,s) : (t,s) ∈ R i j

The upper Darboux Δ-sum U( f ,P) and the lower Darboux Δ-sum L( f ,P) of f with

re-spect toP are defined by

U( f ,P) =n

i =1

k



j =1

M i j

t i − t i −1



s j − s j −1



,

L( f ,P) =

n



i =1

k



j =1

m i j



t i − t i −1



s j − s j −1



.

(2.9)

Note that

U( f ,P) ≤n

i =1

k



j =1

M

t i − t i −1



s j − s j −1



= M(b − a)(d − c) (2.10)

and likewiseL( f ,P) ≥ m(b − a)(d − c) so that

m(b − a)(d − c) ≤ L( f ,P) ≤ U( f ,P) ≤ M(b − a)(d − c). (2.11)

The upper Darboux Δ-integral U( f ) of f over R and the lower Darboux Δ-integral L( f )

of f over R are defined by

U( f ) =inf

U( f ,P) : P ∈ ᏼ(R), L( f ) =sup

L( f ,P) : P ∈ ᏼ(R). (2.12)

In view of (2.11),U( f ) and L( f ) are finite real numbers It can be shown that L( f ) ≤ U( f ).

Definition 2.1 The function f is called Δ-integrable (or delta integrable) over R provided L( f ) = U( f ) In this case,

R f (t,s)Δ1tΔ2s is used to denote this common value This integral is called the Riemann Δ-integral.

We need the following auxiliary result The proof can be found in [5, Lemma 5.7]

Lemma 2.2 For every δ > 0 there exists at least one partition P1∈ ᏼ([a,b)) generated by a set



t0,t1, ,t n

⊂[a,b], where a = t0< t1< ··· < t n = b, (2.13)

such that for each i ∈ {1, 2, ,n } either

or

t i − t i −1> δ, ρ1



t i

Definition 2.3 The set of all P1∈ ᏼ([a,b)) that possess the property indicated inLemma 2.2is denoted byᏼδ([a,b)) Similarly, ᏼ δ([c,d)) is defined Further, the set of all P ∈ ᏼ(R)

such that

P = P1× P2, whereP1∈ᏼδ

[a,b)

,P2∈ᏼδ

[c,d)

(2.16)

is denoted byᏼδ(R).

The following is a Cauchy criterion for Riemann delta integrability (see [3, Theorem 2.11])

Theorem 2.4 A bounded function f on R is Δ-integrable if and only if for each ε > 0 there exists δ > 0 such that

P ∈ᏼδ(R) implies U( f ,P) − L( f ,P) < ε. (2.17)

Remark 2.5 In the two-variable time scales case, four types of integrals can be defined:

(i)ΔΔ-integral over [a,b) ×[c,d), which is introduced by using partitions consisting

of subrectangles of the form [α,β) ×[γ,δ);

(ii)∇∇-integral over ( a,b] ×(c,d], which is defined by using partitions consisting of

subrectangles of the form (α,β] ×(γ,δ];

(iii)Δ∇-integral over [a,b) ×(c,d], which is defined by using partitions consisting of

subrectangles of the form [α,β) ×(γ,δ];

(iv)∇Δ-integral over (a,b] ×[c,d), which is defined by using partitions consisting of

subrectangles of the form (α,β] ×[γ,δ).

For brevity the first integral we call simply aΔ-integral, and in this paper we are dealing mainly with suchΔ-integrals

Fori =1, 2 let us introduce the setT 0

i which is obtained fromTiby removing a possible finite maximal point ofTi, that is, ifTi has a finite maximumt ∗, thenT 0

i = T i \ { t ∗ },

otherwiseT 0

i = T i Briefly we will writeT 0

i = T i \ {maxTi } Evidently, for every point t ∈

T 0

i there exists an interval of the form [α,β) ⊂ T i(withα,β ∈ T iandα < β) that contains

the pointt.

Definition 2.6 Let E ⊂ T0× T0be a bounded set and let f be a bounded function defined

on the setE Let R =[a,b) ×[c,d) ⊂ T1× T2be a rectangle containingE (obviously such

a rectangleR exists) and define F on R as follows:

F(t,s) =

⎧

⎨

⎩

f (t,s) if (t,s) ∈ E,

Then f is said to be Riemann Δ-integrable over E if F is Riemann Δ-integrable over R in

the sense ofDefinition 2.1, and

E f (t,s)Δ1tΔ2s =

R F(t,s)Δ1tΔ2s. (2.19)

IfE is an arbitrary bounded subset ofT 0× T0, then even constant functions need not be RiemannΔ-integrable over E In connection with this we introduce the so-called Jordan Δ-measurable subsets ofT 0× T0 The definition makes use of theΔ-boundary of

a setE ⊂ T1× T2 First, we recall the definition of the usual boundary of a set as given

in [3]

Definition 2.7 Let E ⊂ T1× T2 A pointx =(t,s) ∈ T1× T2is called a boundary point of E

if every open (two-dimensional) ballB(x;r) = { y ∈ T1× T2:d(x, y) < r }of radiusr and

centerx contains at least one point of E and at least one point of (T 1× T2)\ E, where d

is the usual Euclidean distance The set of all boundary points ofE is called the boundary

ofE and is denoted by ∂E.

Definition 2.8 Let E ⊂ T1× T2 A pointx =(t,s) ∈ T1× T2is called aΔ-boundary point

of E if x ∈ T0× T0 and every rectangle of the form [t,t )×[s,s )⊂ T1× T2 witht ∈

T 1,t > t, and s ∈ T2,s > s, contains at least one point of E and at least one point of

(T 1× T2)\ E The set of all Δ-boundary points of E is called the Δ-boundary of E, and it

is denoted by∂ΔE.

Definition 2.9 A point (t0,s0)∈ T0× T0is calledΔ-dense if every rectangle of the form

V =[t0,t) ×[s0,s) ⊂ T1× T2 witht ∈ T1,t > t0, ands ∈ T2,s > s0, contains at least one point ofT 1× T2distinct from (t0,s0) Otherwise the point (t0,s0) is calledΔ-scattered.

Note that in the single variable caseΔ-dense points are precisely the right-dense points, andΔ-scattered points are precisely the right-scattered points Also, a point (t0,s0)∈ T0×

T 0isΔ-dense if and only if at least one of t0ands0is right dense inT 1andT 2, respectively Obviously, eachΔ-boundary point of E is a boundary point of E, but the converse is

not necessarily true Also, eachΔ-boundary point of E must belong toT 0× T0and must

be aΔ-dense point inT 1× T2

Example 2.10 We consider the following examples.

(i) For arbitrary time scalesT 1andT 2, the rectangle of the formE =[a,b) ×[c,d) ⊂

T 1× T2, wherea,b ∈ T1,a < b, and c,d ∈ T2,c < d, has no Δ-boundary point,

that is,∂ΔE = ∅.

(ii) IfT 1= T2= Z, then any set E ⊂ Z × Zhas no boundary as well as noΔ-boundary points

(iii) LetT 1= T2= Randa,b,c,d ∈ Rwitha < b and c < d Let us set

E1=[a,b) ×[c,d), E2=(a,b] ×(c,d], E3=[a,b] ×[c,d]. (2.20)

Then all three rectanglesE1,E2, andE3have the boundary consisting of the union

of all four sides of the rectangle Moreover,∂ΔE1 is empty,∂ΔE2 consists of the union of all four sides of the rectangleE2, and∂ΔE3consists of the union of the right and upper sides ofE3

(iv) LetT 1= T2=[0, 1]∪ {2}, where [0, 1] is the real number interval, and let E =

[0, 1)×[0, 1) Then the boundary∂E of E consists of the union of the right and

upper sides of the rectangleE whereas ∂ΔE = ∅.

(v) LetT 1= T2=[0, 1]∪ {( n + 1)/n : n ∈ N}, where [0, 1] is the real number

inter-val, and letE =[0, 1]×[0, 1] Then the boundary∂E as well as the Δ-boundary

∂ΔE of E coincides with the union of the right and upper sides of E.

Definition 2.11 Let E ⊂ T0

1× T0

2be a bounded set and let∂ΔE be its Δ-boundary Let R =

[a,b) ×[c,d) be a rectangle inT 1× T2such thatE ∪ ∂ΔE ⊂ R Further, let ᏼ(R) denote

the set of allΔ-partitions of R of type (2.5) and (2.6) For everyP ∈ ᏼ(R) define J ∗(E,P)

to be the sum of the areas (for a rectangleV =[α,β) ×[γ,δ) ⊂ T1× T2, its “area” is the numberm(V) =(β − α)(δ − γ)) of those subrectangles of P which are entirely contained

inE, and let J ∗(E,P) be the sum of the areas of those subrectangles of P each of which

contains at least one point ofE ∪ ∂ΔE The numbers

J∗(E) =sup

J∗(E,P) : P ∈ ᏼ(R), J ∗(E) =inf

J ∗(E,P) : P ∈ ᏼ(R) (2.21)

are called the (two-dimensional) inner and outer Jordan Δ-measure of E, respectively The

setE is said to be Jordan Δ-measurable if J∗(E) = J ∗(E), in which case this common value

is called the Jordan Δ-measure of E, denoted by J(E) The empty set is assumed to have

JordanΔ-measure zero

It is easy to verify that 0≤ J∗(E) ≤ J ∗(E) always and that

J ∗(E) − J∗(E) = J ∗

∂ΔE

HenceE is Jordan Δ-measurable if and only if its Δ-boundary ∂ΔE has Jordan Δ-measure

zero

Note that every rectangle R =[a,b) ×[c,d) ⊂ T1× T2, where a,b ∈ T1,a < b, and c,d ∈ T2,c < d, is Jordan Δ-measurable with Jordan Δ-measure J(R) =(b − a)(d − c).

Indeed, it is easily seen that theΔ-boundary of R is empty, and therefore it has Jordan

Δ-measure zero

For each pointx =(t,s) ∈ T0× T0, the single point set{ x }is JordanΔ-measurable, and its Jordan measure is given by

J

{ x }=σ1(t) − t

σ2(s) − s

= μ1(t)μ2(s). (2.23)

Example 2.12 LetT 1= T2= R(orT 1= ZandT 2= R), let a < b be rational numbers in

T 1, and letc < d be rational numbers inT 2 Further, let [a,b] be the interval inT 1, let [c,d] be the interval inT 2, and letE be the set of all points of [a,b] ×[c,d] with rational

coordinates ThenE is not Jordan Δ-measurable The inner Jordan Δ-measure of E is 0

(there is no nonemptyrectangle entirely contained in E), while the outer Jordan

Δ-measure ofE is equal to (σ1(b) − a)(d − c) The Δ-boundary of E coincides with E itself.

IfE ⊂ T0× T0is any bounded JordanΔ-measurable set, then the integralE1Δ1tΔ2s

exists, and we have

In the caseT 1= T2= Z, for any bounded set E ⊂ Z × Zwe have∂ΔE =0, and therefore

E is Jordan Δ-measurable In this case, for any function f : E → R, we have

E f (t,s)Δ1tΔ2s = 

(t,s) ∈ E

and the JordanΔ-measure of E coincides with the number of points of E.

Remark 2.13 Suppose thatT 1has a finite maximumτ0(1) Since by definitionσ1(τ0(1))=

τ0(1), it is reasonable in view of (2.23) to assume thatJ( { x }) =0 for any pointx of the form

x =(τ0(1),s), where s ∈ T2 Also, ifT 2has a finite maximumτ0(2), then we can assume that

J( { y }) =0 for any pointy of the form y =(t,τ0(2)), wheret ∈ T1

3 Multiple Lebesgue integration

First, for the convenience of the reader, we briefly describe the Carath´eodory construction

of a Lebesgue measure in an abstract setting (see [1,6–8]) LetX be an arbitrary set A

collection (family)᏿ of subsets of X is called a semiring if

(i)∅ ∈᏿;

(ii)A,B ∈ ᏿, then A ∩ B ∈᏿;

(iii)A,B ∈ ᏿ and B ⊂ A, then A \ B can be represented as a finite union

A \ B = n

k =1

of pairwise disjoint setsC k ∈᏿

A (set) functionm : ᏿ →[0,∞] whose domain is᏿ and whose values belong to the ex-tended real half-line [0,∞] is said to be a measure on᏿ if

(i)m( ∅) =0;

(ii)m is (finitely) additive in the sense that if A ∈ ᏿ such that A = ∪ n

i =1A i, where

A1, ,A n ∈᏿ are pairwise disjoint, then

m(A) =

n



i =1

m

A i



A measurem with domain of definition ᏿ is said to be countably additive (or σ-additive)

if, for every sequence{ A i }of disjoint sets in᏿ whose union is also in ᏿, we have

m

∞

i =1

A i



=

∞



i =1

m

A i

Letᏼ(X) be the collection of all subsets of X, ᏿ a semiring of subsets of X, and m : ᏿ →

[0,∞] aσ-additive measure of ᏿ Define the set function

as follows LetE be any subset of X If there exists at least one finite or countable system

of setsV i ∈ ᏿, i =1, 2, , such that E ⊂ ∪ i V i, then we put

m ∗(E) =inf

i

m

V i



where the infimum is taken over all coverings ofE by a finite or countable system of sets

V i ∈ ᏿ If there is no such covering of E, then we put m ∗(E) = ∞ The set function m ∗ defined above is called an outer measure on ᏼ(X) (or on X) generated by the pair (᏿,m).

The outer measurem ∗is defined for eachE ⊂ X, however, it cannot be taken as a measure

onX because it is not additive in general In order to guarantee the additivity of m ∗we must restrictm ∗to the collection of the so-called measurable subsets ofX A subset A of

X is said to be m ∗ -measurable (or measurable with respect to m ∗) if

m ∗(E) = m ∗(E ∩ A) + m ∗

E ∩ A C

whereA C = X \ A denotes the complement of the set A A fundamental fact of measure

theory is that (see [6, Theorem 1.3.4]) the familyM(m ∗) of allm ∗-measurable subsets of

X is a σ-algebra (i.e.,M(m ∗) containsX and is closed under the formation of countable

unions and of complements) and the restriction ofm ∗toM(m ∗), which we denote by

μ, is a σ-additive measure onM(m ∗) We have᏿⊂ M( m ∗) andμ(V) = m(V) for each

V ∈ ᏿ The measure μ is called the Carath´eodory extension of the original measure m

defined on the semiring᏿ The measure μ obtained in this way is also called the Lebesgue measure on X generated by the pair (᏿,m) Note that the main difference between the

Jor-dan and Lebesgue-Carath´eodory constructions of measure of a set is that the JorJor-dan con-struction makes use only of finite coverings of the set by some “elementary” sets whereas the Lebesgue-Carath´eodory construction together with the finite coverings admits count-able coverings as well Due to this fact the notion of the Lebesgue measure generalizes the notion of the Jordan measure

Passing now on to time scales, letn ∈ Nbe fixed For eachi ∈ {1, ,n }, letTidenote

a time scale and letσ i,ρ i, andΔidenote the forward jump operator, the backward jump operator, and the delta differentiation operator onTi, respectively Let us set

Λn = T1× ··· × T n =t =t1, ,t n

:t i ∈ T i, 1≤ i ≤ n

We callΛnann-dimensional time scale The set Λ nis a complete metric space with the metricd defined by

d(t,s) =

n

i =1

t i − s i 2

 1/2

Denote byᏲ the collection of all rectangular parallelepipeds in Λnof the form

V =a1,b1 

× ··· ×a n,b n

=t =t1, ,t n

∈Λn:a i ≤ t i < b i, 1≤ i ≤ n

(3.9) with a =(a1, ,a n), b =(b1, ,b n)∈Λn, and a i ≤ b i for all 1≤ i ≤ n If a i = b i for some values of i, then V is understood to be the empty set We will call V also an

(n-dimensional) left-closed and right-open interval in Λ n and denote it by [a,b) Let

m : Ᏺ →[0,∞) be the set function that assigns to each parallelepipedV =[a,b) its

vol-ume:

m(V) =

n



i =1



b i − a i



Then it is not difficult to verify that Ᏺ is a semiring of subsets of Λnandm is a σ-additive

measure onᏲ By μΔwe denote the Carath´eodory extension of the measurem defined on

the semiringᏲ and call μΔthe Lebesgue Δ-measure on Λ n Them ∗-measurable subsets

of Λn will be called Δ-measurable sets and m ∗-measurable functions will be called

Δ-measurable functions.

Theorem 3.1 LetT 0

i = T i \ {maxTi } For each point

t =t1, ,t n

∈ T0× ··· × T0

the single-point set { t } is Δ-measurable, and its Δ-measure is given by

μΔ 

{ t }=n

i =1



σ i

t i

− t i

=n

i =1

μ i

t i

Proof If t i < σ i(t i) for all 1≤ i ≤ n, then

{ t } =t1,σ1



t1



× ··· ×t n,σ n

t n

Therefore{ t }isΔ-measurable and

μΔ 

{ t }= m

t1,σ1



t1



× ··· ×t n,σ n



t n



=

n



i =1



σ i



t i



− t i



which is the desired result Further consider the case whent i = σ i(t i) for some values

ofi ∈ {1, ,n }andt i < σ i(t i) for the other values ofi To illustrate the proof, suppose

t i = σ i(t i) for 1≤ i ≤ n −1 andt n < σ n(t n) In this case for eachi ∈ {1, ,n −1}there exists a decreasing sequence{ t i(k) } k ∈Nof points of Ti such thatt(i k) > t iandt(i k) → t i as

k → ∞ Consider the parallelepipeds inΛn,

V(k) =t1,t1(k)



× ··· ×t n −1,t n(k) −1



×t n,σ n

t n

fork ∈ N (3.15) Then

V(1)⊃ V(2)⊃ ···, { t } =

∞



k =1

Hence{ t }isΔ-measurable as a countable intersection of Δ-measurable sets, and by the

continuity property of theσ-additive measure μΔ, we have

μΔ 

{ t }=lim

k →∞ μΔ 

V(k)

=lim

k →∞



t1(k) − t1



···t n(k) −1− t n −1



σ n

t n

− t n

=0, (3.17)

Example 3.2 In the caseT 1= ··· = T n = R, the measure μΔcoincides with the ordinary Lebesgue measure onRn In the caseT 1= ··· = T n = Z, for any E ⊂ Z n,μΔ(E) coincides

with the number of points of the setE.

Example 3.3 The set E given above inExample 2.12is LebesgueΔ-measurable and its (two-dimensional) LebesgueΔ-measure is equal to zero

Having theσ-additive measure μΔonΛn, we possess the corresponding integration theory for functionsf : E ⊂Λn → R, according to the general Lebesgue integration theory

(see, e.g., [6]) The Lebesgue integral associated with the measureμΔonΛnis called the

Lebesgue Δ-integral For a Δ-measurable set E ⊂Λnand aΔ-measurable function f : E →

R, the correspondingΔ-integral of f over E will be denoted by

E f

t1, ,t n



Δ1t1···Δn t n,

E f (t)Δt, or

E f dμΔ. (3.18)

So all theorems of the general Lebesgue integration theory, including the Lebesgue dom-inated convergence theorem, hold also for LebesgueΔ-integrals on Λn Finally, we com-pare the LebesgueΔ-integral with the Riemann Δ-integral

Theorem 3.4 Let V =[a,b) be a rectangular parallelepiped in Λ n of the form ( 3.9 ) and let

f be a bounded real-valued function on V If f is Riemann Δ-integrable over V, then f is Lebesgue Δ-integrable over V, and

R

V f (t)Δt = L

where R and L indicate the Riemann and Lebesgue Δ-integrals, respectively.

Proof Suppose that f is Riemann Δ-integrable over V =[a,b) Then, byTheorem 2.4, for eachk ∈ Nwe can chooseδ k > 0 (with δ k →0 ask → ∞) and a Δ-partition P(k) = { R(i k) } N(k)

i =1 ofV such that P(k) ∈ᏼδ k(V) and U( f ,P(k))− L( f ,P(k))< 1/k Hence

lim

k →∞ L

f ,P(k)

=lim

k →∞ U

f ,P(k)

= R

By replacing the partitionsP(k)with finer partitions if necessary, we can assume that for eachk ∈ Nthe partitionP(k+1)is a refinement of the partitionP(k) Let us set

m(i k) =inf

f (t) : t ∈ R(i k)

, M(i k) =sup

f (t) : t ∈ R(i k)

(3.21) fori =1, 2, ,N(k), and define sequences { ϕ k } k ∈Nand{ φ k } k ∈Nof functions onV by

letting

ϕ k(t) ≡ m(i k), φ k(t) ≡ M i(k) fort ∈ R(i k), 1≤ i ≤ N(k). (3.22)

(see, e.g., [6]) The Lebesgue. .. the general Lebesgue integration theory, including the Lebesgue dom-inated convergence theorem, hold also for Lebesgue? ?-integrals on Λn Finally, we com-pare the Lebesgue? ?-integral... T1

3 Multiple Lebesgue integration< /b>

First, for the convenience of the reader, we briefly describe the Carath´eodory construction

of a Lebesgue measure in an abstract