MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES

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1 MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES MARTIN BOHNER AND GUSEIN SH. GUSEINO Receved 26 January 2006; Revsed 17 Aprl 2006; Accepted 18 Aprl 2006 We study the process of multple Lebesgue ntegraton on tme scales. The relatonshp of the Remann and the Lebesgue multple ntegrals s nvestgated. Copyrght 2006 M. Bohner and G. Sh. Gusenov. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1. Introducton Dfferental and ntegral calculus on tme scales allows to develop a theory of dynamc equatons n order to unfy and extend the usual dfferental equatons and dfference equatons. For sngle varable dfferental and ntegral calculus on tme scales, we refer the reader to the textbooks [4, 5] and the references gven theren. Multvarable calculus on tme scales was developed by the authors[2, 3]. In [3], we presented the process of Remann multple delta (nabla and mxed types ntegraton on tme scales. In the present paper, we ntroduce the defntons of Lebesgue mult-dmensonal delta (nabla and mxed types measures and ntegrals on tme scales. A comparson of the Lebesgue multple delta ntegral wth the Remann multple delta ntegral s gven. Besde ths ntroductory secton, ths paper conssts of two sectons. In Secton 2, followng [3], we gve the Darboux defnton of the Remann multple delta ntegral and present some needed facts connected to t. The man part of ths paper s Secton 3.There, a bref descrpton of the Carathéodory constructon of a Lebesgue measure n an abstract settng s gven. Then the Lebesgue mult-dmensonal delta measure on tme scales s ntroduced and the Lebesgue delta measure of any sngle-pont set s calculated. When we have a measure, ntegraton theory s avalable accordng to the well-known general scheme of the Lebesgue ntegraton process. Fnally, we compare the Lebesgue multple delta ntegral wth the Remann multple delta ntegral. We ndcate also a way to defne, along wth the Lebesgue mult-dmensonal delta measure, the nabla and mxed types Lebesgue mult-dmensonal measures on tme scales. Hndaw Publshng Corporaton AdvancesnDfference Equatons olume 2006, Artcle ID 26391, Pages 1 12 DOI /ADE/2006/26391

2 2 Multple Lebesgue ntegraton on tme scales 2. Multple Remann ntegraton In ths secton, followng [3], we gve the defnton of the multple Remann ntegral on tme scales over arbtrary bounded regons as we wll compare n the next secton the Remann ntegral wth the Lebesgue ntegral ntroduced theren. For convenence, we present the exposton for functons of two ndependent varables. Let T 1 and T 2 be two tme scales. For = 1,2, let σ, ρ,andδ denote the forward jump operator, the backward jump operator, and the delta dfferentaton operator, respectvely, on T.Supposea<bare ponts n T 1, c<dare ponts n T 2,[a,b sthehalfclosed bounded nterval n T 1,and[c,d s the half-closed bounded nterval n T 2.Letus ntroduce a rectangle (or delta rectangle n T 1 T 2 by R = [a,b [c,d = { (t,s:t [a,b, s [c,d }. (2.1 Frst we defne Remann ntegrals over rectangles of the type gven n (2.1. Let { t0,t 1,...,t n } [a,b], where a = t0 <t 1 < <t n = b, { s0,s 1,...,s k } [c,d], where c = s0 <s 1 < <s k = d. (2.2 The numbers n and k may be arbtrary postve ntegers. We call the collecton of ntervals P 1 = {[ } t 1,t :1 n (2.3 a Δ-partton (or delta parttonof[a,b anddenotethe set of allδ-parttons of [a,bby ([a,b. Smlarly, the collecton of ntervals P 2 = {[ } s j 1,s j :1 j k (2.4 s called a Δ-partton of [c,d and the set of all Δ-parttons of [c,d s denoted by ([c,d. Let us set We call the collecton R j = [ t 1,t [ sj 1,s j, where1 n, 1 j k. (2.5 P = { R j :1 n,1 j k } (2.6 a Δ-partton of R, generated by the Δ-parttons P 1 and P 2 of [a,band[c,d, respectvely, and wrte P = P 1 P 2.TherectanglesR j,1 n,1 j k, are called the subrectangles of the partton P. The set of all Δ-parttons of R s denoted by (R. Let f : R R be a bounded functon. We set and for 1 n,1 j k, M = sup { f (t,s:(t,s R }, m = nf { f (t,s:(t,s R } (2.7 M j = sup { f (t,s:(t,s R j }, mj = nf { f (t,s:(t,s R j }. (2.8

3 M. Bohner and G. Sh. Gusenov 3 The upper Darboux Δ-sum U( f,p and the lower Darboux Δ-sum L( f,p of f wth respect to P are defned by Note that U( f,p U( f,p = L( f,p = n j=1 n j=1 n j=1 and lkewse L( f,p m(b a(d csothat k ( ( M j t t 1 sj s j 1, (2.9 k ( ( m j t t 1 sj s j 1. k M ( ( t t 1 sj s j 1 = M(b a(d c (2.10 m(b a(d c L( f,p U( f,p M(b a(d c. (2.11 The upper Darboux Δ-ntegral U( f of f over R and the lower Darboux Δ-ntegral L( f of f over R are defned by U( f = nf { U( f,p:p (R }, L( f = sup { L( f,p:p (R }. (2.12 In vew of (2.11, U( f andl( f are fnte real numbers. It can be shown that L( f U( f. Defnton 2.1. The functon f s called Δ-ntegrable (or delta ntegrableover R provded L( f = U( f. In ths case, R f (t,sδ 1 tδ 2 s s used to denote ths common value. Ths ntegral s called the Remann Δ-ntegral. We need the followngauxlaryresult.the proof can be foundn [5, Lemma 5.7]. Lemma 2.2. For every δ>0thereexstsatleastoneparttonp 1 ([a,b generated by a set { } t0,t 1,...,t n [a,b], where a = t0 <t 1 < <t n = b, (2.13 such that for each {1,2,...,n} ether or t t 1 δ (2.14 t t 1 >δ, ρ 1 ( t = t 1. (2.15 Defnton 2.3. The set of all P 1 ([a,b that possess the property ndcated n Lemma 2.2 s denoted by δ ([a,b. Smlarly, δ ([c,d s defned. Further, the set of all P (R

4 4 Multple Lebesgue ntegraton on tme scales such that P = P 1 P 2, wherep 1 δ ( [a,b, P2 δ ( [c,d (2.16 s denoted by δ (R. The followng s a Cauchy crteron for Remann delta ntegrablty (see [3, Theorem 2.11]. Theorem 2.4. A bounded functon f on R s Δ-ntegrable f and only f for each ε>0 there exsts δ>0 such that P δ (R mples U( f,p L( f,p <ε. (2.17 Remark 2.5. In the two-varable tme scales case, four types of ntegrals can be defned: ( ΔΔ-ntegral over [a,b [c,d, whch s ntroduced by usng parttons consstng of subrectangles of the form [α,β [γ,δ; ( -ntegral over (a,b] (c,d], whch s defned by usng parttons consstng of subrectangles of the form (α,β] (γ,δ]; ( Δ -ntegral over [a,b (c,d], whch s defned by usng parttons consstng of subrectangles of the form [α,β (γ,δ]; (v Δ-ntegral over (a,b] [c,d, whch s defned by usng parttons consstng of subrectangles of the form (α,β] [γ,δ. For brevty the frst ntegral we call smply a Δ-ntegral, and n ths paper we are dealng manly wth such Δ-ntegrals. For = 1,2 let us ntroduce the set T 0 whchsobtanedfromt by removng a possble fnte maxmal pont of T, that s, f T has a fnte maxmum t,thent 0 = T \{t }, otherwse T 0 = T. Brefly we wll wrte T 0 = T \{maxt }. Evdently, for every pont t T 0 there exsts an nterval of the form [α,β T (wth α,β T and α<β that contans the pont t. Defnton 2.6. Let E T 0 1 T 0 2 be a bounded set and let f be a bounded functon defned on the set E.LetR = [a,b [c,d T 1 T 2 be a rectangle contanng E (obvously such a rectangle R exsts and defne F on R as follows: f (t,s f(t,s E, F(t,s = 0 f(t,s R \ E. (2.18 Then f s sad to be Remann Δ-ntegrable over E f F s Remann Δ-ntegrable over R n the sense of Defnton 2.1,and f (t,sδ 1 tδ 2 s = F(t,sΔ 1 tδ 2 s. (2.19 E If E s an arbtrary bounded subset of T 0 1 T 0 2, then even constant functons need not be Remann Δ-ntegrable over E. In connecton wth ths we ntroduce the so-called Jordan Δ-measurable subsets of T 0 1 T 0 2. The defnton makes use of the Δ-boundary of R

5 M. Bohner and G. Sh. Gusenov 5 asete T 1 T 2. Frst, we recall the defnton of the usual boundary of a set as gven n [3]. Defnton 2.7. Let E T 1 T 2.Apontx = (t,s T 1 T 2 s called a boundary pont of E f every open (two-dmensonal ball B(x;r ={y T 1 T 2 : d(x, y <r} of radus r and center x contans at least one pont of E and at least one pont of (T 1 T 2 \ E, whered s the usual Eucldean dstance. The set of all boundary ponts of E s called the boundary of E and s denoted by E. Defnton 2.8. Let E T 1 T 2.Apontx = (t,s T 1 T 2 s called a Δ-boundary pont of E f x T 0 1 T 0 2 and every rectangle of the form [t,t [s,s T 1 T 2 wth t T 1, t >t,ands T 2, s >s, contans at least one pont of E and at least one pont of (T 1 T 2 \ E. The set of all Δ-boundary ponts of E s called the Δ-boundary of E,andt s denoted by Δ E. Defnton 2.9. Apont(t 0,s 0 T 0 1 T 0 2 s called Δ-dense f every rectangle of the form = [t 0,t [s 0,s T 1 T 2 wth t T 1, t>t 0,ands T 2, s>s 0, contans at least one pont of T 1 T 2 dstnct from (t 0,s 0. Otherwse the pont (t 0,s 0 scalledδ-scattered. Note that n the sngle varable case Δ-dense ponts are precsely the rght-dense ponts, and Δ-scattered ponts are precsely the rght-scattered ponts. Also, a pont (t 0,s 0 T 0 1 T 0 2 s Δ-dense f and only f at least one of t 0 and s 0 s rght dense n T 1 and T 2, respectvely. Obvously, each Δ-boundary pont of E s a boundary pont of E, buttheconverses not necessarly true. Also, each Δ-boundary pont of E must belong to T 0 1 T 0 2 and must be a Δ-dense pont n T 1 T 2. Example We consder the followng examples. ( For arbtrary tme scales T 1 and T 2, the rectangle of the form E = [a,b [c,d T 1 T 2,wherea,b T 1, a<b,andc,d T 2, c<d,hasnoδ-boundary pont, that s, Δ E =. ( If T 1 = T 2 = Z, then any set E Z Z hasnoboundaryaswellasnoδ-boundary ponts. ( Let T 1 = T 2 = R and a,b,c,d R wth a<band c<d.letusset E 1 = [a,b [c,d, E 2 = (a,b] (c,d], E 3 = [a,b] [c,d]. (2.20 Then all three rectangles E 1, E 2,andE 3 have the boundary consstng of the unon of all four sdes of the rectangle. Moreover, Δ E 1 s empty, Δ E 2 conssts of the unon of all four sdes of the rectangle E 2,and Δ E 3 conssts of the unon of the rght and upper sdes of E 3. (v Let T 1 = T 2 = [0,1] {2}, where [0,1] s the real number nterval, and let E = [0,1 [0,1. Then the boundary E of E conssts of the unon of the rght and upper sdes of the rectangle E whereas Δ E =. (v Let T 1 = T 2 = [0,1] {(n +1/n : n N}, where [0,1] s the real number nterval, and let E = [0,1] [0,1]. Then the boundary E as well as the Δ-boundary Δ E of E concdes wth the unon of the rght and upper sdes of E.

6 6 Multple Lebesgue ntegraton on tme scales Defnton Let E T 0 1 T 0 2 be a bounded set and let Δ E be ts Δ-boundary. Let R = [a,b [c,d bearectanglent 1 T 2 such that E Δ E R. Further,let (R denote the set of all Δ-parttons of R of type (2.5and(2.6. For every P (RdefneJ (E,P to be the sum of the areas (for a rectangle = [α,β [γ,δ T 1 T 2, ts area s the number m( = (β α(δ γ of those subrectangles of P whch are entrely contaned n E, andletj (E,P be the sum of the areas of those subrectangles of P each of whch contans at least one pont of E Δ E.Thenumbers J (E = sup { J (E,P:P (R }, J (E = nf { J (E,P:P (R } (2.21 are called the (two-dmensonal nner and outer Jordan Δ-measure of E, respectvely. The set E s sad to be Jordan Δ-measurable f J (E = J (E, n whch case ths common value s called the Jordan Δ-measure of E, denoted by J(E. The empty set s assumed to have Jordan Δ-measure zero. It s easy to verfy that 0 J (E J (E always and that J (E J (E = J ( Δ E. (2.22 Hence E s Jordan Δ-measurable f and only f ts Δ-boundary Δ E has Jordan Δ-measure zero. Note that every rectangle R = [a,b [c,d T 1 T 2,wherea,b T 1, a<b,and c,d T 2, c<d,sjordanδ-measurable wth Jordan Δ-measure J(R = (b a(d c. Indeed, t s easly seen that the Δ-boundary of R s empty, and therefore t has Jordan Δ-measure zero. For each pont x = (t,s T 0 1 T 0 2, the sngle pont set {x} s Jordan Δ-measurable, and ts Jordan measure s gven by J ( {x} = ( σ 1 (t t ( σ 2 (s s = μ 1 (tμ 2 (s. (2.23 Example Let T 1 = T 2 = R (or T 1 = Z and T 2 = R, let a<bbe ratonal numbers n T 1,andletc<dbe ratonal numbers n T 2.Further,let[a,b] be the nterval n T 1,let [c,d] be the nterval n T 2,andletE be the set of all ponts of [a,b] [c,d] wth ratonal coordnates. Then E s not Jordan Δ-measurable. The nner Jordan Δ-measure of E s 0 (there s no nonempty Δ-rectangle entrely contaned n E, whle the outer Jordan Δ- measure of E s equal to (σ 1 (b a(d c. The Δ-boundary of E concdes wth E tself. If E T 0 1 T 0 2 s any bounded Jordan Δ-measurable set, then the ntegral E 1Δ 1 tδ 2 s exsts, and we have 1Δ 1 tδ 2 s = J(E. (2.24 E

7 M. Bohner and G. Sh. Gusenov 7 In the case T 1 = T 2 = Z, for any bounded set E Z Z we have Δ E = 0, and therefore E s Jordan Δ-measurable. In ths case, for any functon f : E R,wehave E f (t,sδ 1 tδ 2 s = (t,s E f (t,s, (2.25 and the Jordan Δ-measure of E concdes wth the number of ponts of E. Remark Suppose that T 1 has a fnte maxmum τ (1 0. Snce by defnton σ 1 (τ (1 0 = τ (1 0, t s reasonable n vew of (2.23toassumethatJ({x} = 0 for any pont x of the form x = (τ (1 0,s, where s T 2.Also,fT 2 has a fnte maxmum τ (2 0, then we can assume that J({y} = 0 for any pont y of the form y = (t,τ (2 0, where t T Multple Lebesgue ntegraton Frst, for the convenence of the reader, we brefly descrbe the Carathéodory constructon of a Lebesgue measure n an abstract settng (see [1, 6 8]. Let X be an arbtrary set. A collecton (famly of subsets of X s called a semrng f ( ; ( A,B,thenA B ; ( A,B and B A,thenA \ B can be represented as a fnte unon A \ B = n C k (3.1 of parwse dsjont sets C k. A (set functon m : [0, ] whose doman s and whose values belong to the extended real half-lne [0, ]ssadtobeameasure on f ( m( = 0; ( m s (fntely addtve n the sense that f A such that A = n A,where A 1,...,A n are parwse dsjont, then m(a = k=1 n m ( A. (3.2 Ameasurem wth doman of defnton s sad to be countably addtve (or σ-addtve f, for every sequence {A } of dsjont sets n whose unon s also n,wehave ( m A = m ( A. (3.3 Let (X be the collecton of all subsets of X, a semrng of subsets of X,andm : [0, ]aσ-addtve measure of. Defne the set functon m : (X [0, ] (3.4

8 8 Multple Lebesgue ntegraton on tme scales as follows. Let E be any subset of X. If there exsts at least one fnte or countable system of sets, = 1,2,...,suchthatE,thenweput m (E = nf m (, (3.5 where the nfmum s taken over all coverngs of E by a fnte or countable system of sets. If there s no such coverng of E, thenweputm (E =. The set functon m defned above s called an outer measure on (X (or on X generated by the par (,m. The outer measure m s defned for each E X, however, t cannot be taken as a measure on X because t s not addtve n general. In order to guarantee the addtvty of m we must restrct m to the collecton of the so-called measurable subsets of X.AsubsetA of X s sad to be m -measurable (or measurable wth respect to m f m (E = m (E A+m ( E A C, E X, (3.6 where A C = X \ A denotes the complement of the set A. A fundamental fact of measure theory s that (see [6, Theorem 1.3.4] the famly M(m ofallm -measurable subsets of X s a σ-algebra (.e., M(m contans X and s closed under the formaton of countable unons and of complements and the restrcton of m to M(m, whch we denote by μ, saσ-addtve measure on M(m. We have M(m andμ( = m( foreach. The measure μ s called the Carathéodory extenson of the orgnal measure m defned on the semrng. The measure μ obtaned n ths way s also called the Lebesgue measure on X generated by the par (,m. Note that the man dfference between the Jordan and Lebesgue-Carathéodory constructons of measure of a set s that the Jordan constructon makes use only of fnte coverngs of the set by some elementary sets whereas the Lebesgue-Carathéodory constructon together wth the fnte coverngs admts countable coverngs as well. Due to ths fact the noton of the Lebesgue measure generalzes the noton of the Jordan measure. Passng now on to tme scales, let n N be fxed. For each {1,...,n},letT denote atmescaleandletσ, ρ,andδ denote the forward jump operator, the backward jump operator, and the delta dfferentaton operator on T, respectvely. Let us set Λ n = T 1 T n = { t = ( t 1,...,t n : t T,1 n }. (3.7 We call Λ n an n-dmensonal tme scale. The set Λ n s a complete metrc space wth the metrc d defned by ( n 1/2 d(t,s = t s 2 for t,s Λ n. (3.8 Denote by the collecton of all rectangular paralleleppeds n Λ n of the form = [ a 1,b 1 [ an,b n = { t = ( t1,...,t n Λ n : a t <b,1 n } (3.9 wth a = (a 1,...,a n, b = (b 1,...,b n Λ n,anda b for all 1 n. Ifa = b for some values of, then s understood to be the empty set. We wll call also an

9 M. Bohner and G. Sh. Gusenov 9 (n-dmensonal left-closed and rght-open nterval n Λ n and denote t by [a,b. Let m : [0, be the set functon that assgns to each parallelepped = [a,b tsvolume: m( = n ( b a. (3.10 Then t s not dffcult to verfy that s a semrng of subsets of Λ n and m s a σ-addtve measure on.byμ Δ we denote the Carathéodory extenson of the measure m defned on the semrng and call μ Δ the Lebesgue Δ-measure on Λ n.them -measurable subsets of Λ n wll be called Δ-measurable sets and m -measurable functons wll be called Δ- measurable functons. Theorem 3.1. Let T 0 = T \{maxt }. For each pont t = ( t 1,...,t n T 0 1 T 0 n (3.11 the sngle-pont set {t} s Δ-measurable, and ts Δ-measures gven by ( n ( ( n ( μ Δ {t} = σ t t = μ t. (3.12 Proof. If t <σ (t forall1 n,then Therefore {t} sδ-measurable and {t}= [ t 1,σ 1 ( t1 [ tn,σ n ( tn. (3.13 ( ([ ( [ ( n ( ( μ Δ {t} = m t1,σ 1 t1 tn,σ n tn = σ t t, (3.14 whch s the desred result. Further consder the case when t = σ (t forsomevalues of {1,...,n} and t <σ (t for the other values of. To llustrate the proof, suppose t = σ (t for1 n 1andt n <σ n (t n. In ths case for each {1,...,n 1} there exsts a decreasng sequence {t (k k. Consder the paralleleppeds n Λ n, Then } k N of ponts of T such that t (k (k = [ t 1,t (k [ 1 tn 1,t (k [ ( n 1 tn,σ n tn >t and t (k t as for k N. (3.15 (1 (2, {t}= (k. (3.16 k=1 Hence {t} s Δ-measurable as a countable ntersecton of Δ-measurable sets, and by the

10 10 Multple Lebesgue ntegraton on tme scales contnuty property of the σ-addtve measure μ Δ,wehave ( ( μ Δ {t} = lm μ Δ (k ( = lm t (k ( 1 t 1 t (k ( ( n 1 t n 1 σn tn tn = 0, k k (3.17 and so (3.12 holds n ths case as well. Example 3.2. In the case T 1 = =T n = R, the measure μ Δ concdes wth the ordnary Lebesgue measure on R n. In the case T 1 = =T n = Z,foranyE Z n, μ Δ (E concdes wth the number of ponts of the set E. Example 3.3. The set E gven above n Example 2.12 s Lebesgue Δ-measurable and ts (two-dmensonal Lebesgue Δ-measure s equal to zero. Havng the σ-addtve measure μ Δ on Λ n, we possess the correspondng ntegraton theory for functons f : E Λ n R, accordng to the general Lebesgue ntegraton theory (see, e.g., [6]. The Lebesgue ntegral assocated wth the measure μ Δ on Λ n s called the Lebesgue Δ-ntegral.ForaΔ-measurable set E Λ n and a Δ-measurable functon f : E R, the correspondng Δ-ntegral of f over E wll be denoted by f ( t 1,...,t n Δ1 t 1 Δ n t n, f (tδt, or fdμ Δ. (3.18 E E So all theorems of the general Lebesgue ntegraton theory, ncludng the Lebesgue domnated convergence theorem, hold also for Lebesgue Δ-ntegrals on Λ n.fnally,wecompare the Lebesgue Δ-ntegral wth the Remann Δ-ntegral. Theorem 3.4. Let = [a,b be a rectangular parallelepped n Λ n of the form (3.9andlet f be a bounded real-valued functon on. If f s Remann Δ-ntegrable over, then f s Lebesgue Δ-ntegrable over,and R f (tδt = L f (tδt, (3.19 where R and L ndcate the Remann and Lebesgue Δ-ntegrals, respectvely. Proof. Suppose that f s Remann Δ-ntegrable over = [a,b. Then, by Theorem 2.4, for each k N we can choose δ k > 0(wthδ k 0ask andaδ-partton P (k = {R (k } N(k of such that P (k δk (andu( f,p (k L( f,p (k < 1/k.Hence lm L ( f,p (k = lm U ( f,p (k = R f (tδt. (3.20 k k By replacng the parttons P (k wth fner parttons f necessary, we can assume that for each k N the partton P (k+1 s a refnement of the partton P (k.letusset m (k = nf { f (t:t R (k }, M (k = sup { f (t:t R (k } E (3.21 for = 1,2,...,N(k, and defne sequences {ϕ k } k N and {φ k } k N of functons on by lettng ϕ k (t m (k, φ k (t M (k for t R (k,1 N(k. (3.22

11 M. Bohner and G. Sh. Gusenov 11 Then {ϕ k } s a nondecreasng and {φ k } s a nonncreasng sequence of smple Δ-measurable functons. For each k N,wehave ϕ k ϕ k+1, φ k φ k+1, (3.23 ϕ k f φ k, (3.24 L ϕ k (tδt = L ( f,p (k, L φ k (tδt = U ( f,p (k. (3.25 Snce f s bounded, the sequences {ϕ k } and {φ k } are bounded by (3.21and(3.22. Addtonally takng the monotoncty (3.23 nto account, we conclude that the lmt functons ϕ(t = lm k ϕ k (t, φ(t = lm k φ k (t fort (3.26 exst and that ϕ(t f (t φ(t, t. (3.27 The functons ϕ and φ are Δ-measurable as lmts (3.26 oftheδ-measurable functons ϕ k and φ k, respectvely. Lebesgue s domnated convergence theorem mples that ϕ and φ are Lebesgue Δ-ntegrable over and lm L ϕ k (tδt = L ϕ(tδt, k lml φ k (tδt = L φ(tδt. (3.28 k Therefore passng on to the lmt n (3.25 ask and takng (3.20 nto account, we obtan L ϕ(tδt = L φ(tδt = R f (tδt. (3.29 From (3.29wefnd [ ] L φ(t ϕ(t Δt = 0. (3.30 Sncenaddtonφ(t ϕ(t 0forallt,(3.30 mples ϕ(t = φ(t Δ-almost everywhere on. (3.31 So (3.27and(3.31 show that f (t = ϕ(t Δ-almost everywhere on. It follows that f s Lebesgue Δ-ntegrable together wth ϕ and that the Lebesgue Δ-ntegrals of f and ϕ over concde. Now from (3.29 we get that the Lebesgue Δ-ntegral of f over concdes wth the Remann Δ-ntegral of f over. Remark 3.5. Let E be a bounded Jordan Δ-measurable set n T 0 1 T 0 n and f : E R a bounded functon. Usng Defnton 2.6 and Theorem 3.4 wegetthatf f s Remann Δ- ntegrable over E,then f s Lebesgue Δ-ntegrable over E, and the values of the ntegrals concde.

12 12 Multple Lebesgue ntegraton on tme scales Remark 3.6. We can defne Lebesgue nabla measure (and hence Lebesgue nabla ntegral on Λ n by replacng the famly of paralleleppeds of the form (3.9 by the famly of paralleleppeds of the form wth the same volume formula = ( a 1,b 1 ] ( an,b n ], (3.32 m( = n ( b a. (3.33 Smlarly we can defne mxed types of Lebesgue measures on Λ n by takng the famles of paralleleppeds n whch one part of the consttuent ntervals of the paralleleppeds s left closed and rght open and the other part s left open and rght closed (see also Remark 2.5. References [1] G. Aslm and G. Sh. Gusenov, Weak semrngs, ω-semrngs, and measures, Bulletn of the Allahabad Mathematcal Socety 14 (1999, [2] M. Bohner and G. Sh. Gusenov, Partal dfferentaton on tme scales, Dynamc Systems and Applcatons 13 (2004, no. 3-4, [3], Multple ntegraton on tme scales, Dynamc Systems and Applcatons 14 (2005, no. 3-4, [4] M. Bohner and A. Peterson, Dynamc Equatons on Tme Scales, Brkhäuser Boston, Massachusetts, [5] M. Bohner and A. Peterson (eds., Advances n Dynamc Equatons on Tme Scales, Brkhäuser Boston, Massachusetts, [6] D. L. Cohn, Measure Theory,Brkhäuser Boston, Massachusetts, [7] P.R.Halmos,Measure Theory, D. an Nostrand, New York, [8] A. N. KolmogorovandS.. Fomīn, Introductory Real Analyss, Dover, New York, Martn Bohner: Department of Mathematcs and Statstcs, Unversty of Mssour-Rolla, Rolla, MO 65401, USA E-mal address: bohner@umr.edu Gusen Sh. Gusenov: Department of Mathematcs, Atlm Unversty, Incek, Ankara, Turkey E-mal address: gusenov@atlm.edu.tr