A GENERAL INTEGRAL RICARDO ESTRADA AND JASSON VINDAS

A GENERAL INTEGRAL RICARDO ESTRADA AND JASSON VINDAS Abstrct. We define n integrl, the distributionl integrl of functions of one rel vrible, tht is more generl thn the Lebesgue nd the Denjoy-Perron-Henstock integrls, nd which lso llows the integrtion of functions with distributionl vlues everywhere or nerly everywhere. Our integrl hs the property tht if f is loclly distributionlly integrble over the rel line nd ψ D (R) is test function, then f ψ is distributionlly integrble, nd the formul f, ψ = (dist) f (x) ψ (x) dx, defines distribution f D (R) tht hs distributionl point vlues lmost everywhere nd ctully f (x) = f (x) lmost everywhere. The indefinite distributionl integrl F (x) = (dist) x f (t) dt corresponds to distribution with point vlues everywhere nd whose distributionl derivtive hs point vlues lmost everywhere equl to f (x). The distributionl integrl is more generl thn the stndrd integrls, but it still hs mny of the useful properties of those stndrd ones, including integrtion by prts formuls, substitution formuls, even for infinite intervls in the Cesàro sense, men vlue theorems, nd convergence theorems. The distributionl integrl stisfies version of Hke s theorem. Unlike generl distributions, loclly distributionlly integrble functions cn be restricted to closed sets nd cn be multiplied by power functions with rel positive exponents. 2000 Mthemtics Subject Clssifiction. Primry 26A39, 46F10. Secondry 26A24, 26A36. Key words nd phrses. Distributions, Lojsiewicz point vlues, distributionl integrtion, generl integrl, non-bsolute integrls. R. Estrd grtefully cknowledges support from NSF, through grnt number 0968448. J. Vinds grtefully cknowledges support by Postdoctorl Fellowship of the Reserch Foundtion Flnders (FWO, Belgium). 1

2 RICARDO ESTRADA AND JASSON VINDAS Contents 1. Introduction 3 2. Preliminries 6 2.1. Spces 6 2.2. Point vlues 7 2.3. The Cesàro behvior of distributions t infinity 8 2.4. Evlutions 9 2.5. Lojsiewicz distributions 10 2.6. Distributionlly regulted functions 12 2.7. Romnovski s lemm 13 2.8. Mesures 13 3. The φ trnsform 14 4. The definite integrl 19 5. The indefinite integrl 23 6. Comprison with other integrls 27 7. Distributions nd integrtion 32 8. Improper integrls 37 9. Convergence theorems 41 10. Chnge of vribles 45 11. Men vlue theorems 48 12. Exmples 50 References 56

1. Introduction In this rticle we construct nd study the properties of generl integrtion opertor tht cn be pplied to functions of one vrible, f : [, b] R = R {, }. We denote this integrl s (1.1) (dist) f (x) dx, nd cll it the distributionl integrl of f. The spce of distributionlly integrble functions is vector spce nd the opertor (1.1) is liner functionl in this spce. The construction gives n integrl with the following properties: 1. Any Denjoy-Perron-Henstock integrble function, nd in prticulr ny Lebesgue integrble function, is lso distributionlly integrble, nd the integrls coincide. If the Denjoy-Perron-Henstock integrl cn be ssigned the vlue + (or ) then the distributionl integrl cn lso be ssigned the vlue + (or ). 2. If distribution f D (R) hs distributionl point vlues (s defined in Subsection 2.2) t ll points of [, b] nd if f (x) = f (x) is the function given by those point vlues, then f is distributionlly integrble over [, b]. 3. If f : R R is function tht is distributionlly integrble over ny compct intervl, nd if ψ D (R) is test function, then the formul (1.2) f (x), ψ (x) = (dist) f (x) ψ (x) dx, defines distribution f D (R). This distribution f hs distributionl point vlues lmost everywhere nd (1.3) f (x) = f (x) (.e.). If we strt with distribution f 0 D (R) tht hs vlues everywhere, then construct the function f given by those vlues, nd then define distribution f D (R) by formul (1.2) then we recover the initil distribution: f = f 0. We cll the integrl generl integrl becuse of property 1, which sys tht it is more generl thn the stndrd integrls. We cll it the distributionl integrl becuse of 2 nd 3, since these properties sy tht functions integrble in this sense re relted to corresponding distributions in very precise fshion. In the sme wy tht loclly integrble Lebesgue functions f give rise to ssocited regulr distributions f, f f, loclly integrble 3

4 RICARDO ESTRADA AND JASSON VINDAS distributionlly functions hve ssocited loclly integrble distributions. Actully Denjoy-Perron-Henstock integrble functions lso hve cnoniclly ssocited distributions [32]. Observe, however, tht for the purposes of this rticle is better to sy tht f nd f re ssocited nd employ different nottions for the function nd the distribution, insted of the stndrd prctice of sying tht f is f. The question of whether distribution cn be ssocited to function or not ws considered in the lecture [17]; understnding tht distributions, in generl, re regulriztions of functions, nd usully not uniquely determined [19] llows one to void common misunderstndings in the formuls used in Mthemticl Physics [25]. Our construction of the integrl is bsed upon chrcteriztion of positive mesures in terms of the properties of the φ trnsform [51, 24, 11, 37], introduced in Section 3. Indeed, in the Theorems 3.4, 3.5, nd 3.7 we give conditions on the pointwise extreme vlues of distribution tht gurntee tht it is positive mesure, nd this llows us to consider the notions of mjor nd minor distributionl pirs nd then, in Definition 4.4, define the distributionl integrl. In Section 4 we show tht the integrl is liner functionl, tht distributionlly integrble functions re finite lmost everywhere nd mesurble, nd tht the integrls of functions tht re equl (.e) coincide. In Section 5 we study the indefinite integrl (1.4) F (x) = (dist) x f (t) dt, of distributionlly integrble function f. We prove tht F is Lojsiewicz function (Definition 2.2), tht is, it hs point vlues everywhere. In generl F will not be continuous but it will be continuous in n verge sense. Other integrtion processes hve discontinuous indefinite integrls [28, Sections 479 482], but they re not even liner opertions. Any Lojsiewicz function hs ssocited unique distribution F, F F, nd thus we my consider its derivtive, f = F. We show tht F hs distributionl vlues lmost everywhere nd tht ctully F (x) = f (x) (.e). This precise sttement of the ide tht f is the derivtive of F lmost everywhere. Lter on, in Section 7, we re ble to show tht f = F is the sme distribution given by (1.2). In Section 6 we show tht our integrl is more generl thn the Lebesgue integrl nd thn the Denjoy-Perron-Henstock integrl. In fct, more generlly, our integrl is cpble of recovering function from its higher order differentil quotients, problem originlly considered by Denjoy in [9]. We lso show tht Lojsiewicz functions nd distributionlly regulted functions [48] re distributionlly integrble,

s re the distributionl derivtives of Lojsiewicz distributions whose point vlues exist nerly everywhere. The reltionship between loclly distributionlly integrble functions nd distributions is studied in Section 7, not only in the spce D (R), but in other spces such s E (R), S (R), or K (R) s well. According to Hke s theorem [27], there re no improper Denjoy- Perron-Henstock integrls over finite intervls, becuse such integrls re ctully ordinry Denjoy-Perron-Henstock integrls. We prove corresponding result, nmely, if f is distributionlly integrble over [, x] for ny x < b, nd if (dist) x f (t) dt hs distributionl limit L s x b, then f is integrble over [, b] nd the integrl is equl to L. We pply this result to show tht if f is distributionlly integrble over [, b] then so re the functions (x ) α (b x) β f (x) for ny rel numbers α > 0 nd β > 0. We prove bounded convergence theorem, monotone convergence theorem, nd version of Ftou s lemm in Section 9. We exmine chnges of vribles in Section 10, showing, in prticulr, tht distributionl integrls become Cesàro type integrls when the chnge sends finite intervl to n infinite one. The three men vlue theorems of integrl clculus re proved in Section 11. In the lst section, Section 12, we provide severl exmples tht illustrte our ides. We give exmples of functions tht re distributionlly integrble but not Denjoy-Perron-Henstock integrble, exmples of distributionlly integrble functions tht re not Lojsiewicz functions, nd exmples of Lojsiewicz functions which re not indefinite integrls. We consider the boundry vlues of the Poisson integrl of distributionlly integrble function. Moreover, we consider the Fourier series of periodic loclly distributionlly integrble functions nd the Fourier trnsform of tempered loclly distributionlly integrble functions. We lso explin why the Cuchy representtion formul (1.5) F (z) = 1 2πi (dist) f (ξ) ξ z dξ, holds for certin functions F nlytic in Im z > 0 whose boundry vlues on R come from loclly integrble distributions (s f (ξ) = ξ 1 e i/ξ, for instnce), nd why such formul does not hold, even in the principl vlue sense, for non distributionlly integrble functions (s f (ξ) = ξ 1, for instnce). There hve been severl studies tht involve distributions nd integrtion. Let us emphsize tht our integrl is method to find the integrl of functions s re, let us sy, the Riemnn or the Denjoy 5

6 RICARDO ESTRADA AND JASSON VINDAS integrls. A completely different question is the integrtion of distributions. Indeed, observe, first of ll, tht the fct tht ny distribution f D (R) hs primitive F D (R), F = f, is trivil. If F hs vlues t x = nd t x = b then we sy tht f is integrble over [, b] nd write [7] (1.6) f (x) dx = F (b) F (). Hence f (x) dx is number. This notion is due to the Polish school [2, 30] nd hs severl pplictions, s in the theory of smpling theorems [58]. On the other hnd, Silv nd Sikorski, independently, used their definitions of the integrl of distributions to write Fourier trnsforms nd convolutions of distributions s integrls [39, 42]. Moreover, severl uthors [4, 44] hve considered the clss of continuously integrble distributions, tht is, those distributions with continuous primitive; observe, however, tht continuously integrble distributions my not hve vlues t ny point, nd thus re not relly functions, in generl. We should point out tht one cn devise simple procedure for the construction of primitives of functions by using the fct tht distributions re known to hve primitives. Indeed, strt with function f, ssocite to it distribution f, construct the distributionl primitive F, tht is, F = f, nd then construct the function F ssocited to F. Then F would be primitive of f. Unfortuntely, this procedure fils, in generl, becuse there is no unique wy to ssign distribution f to given function f, s follows from the Theorem 7.1. Interestingly, however, it does work sometimes, s we show, for instnce, for Lojsiewicz functions, becuse in this cse ll the ssocitions re unique [30]. 2. Preliminries In this section we hve collected severl importnt ides tht will ply role in our construction of generl distributionl integrl. 2.1. Spces. We use the term smooth function to men C function. The Schwrtz spces of test functions D, E, nd S nd the corresponding spces of distributions re well known [2, 29, 41, 43, 55]. Recll tht E consists of ll smooth functions, while D nd S stnd, respectively, for the spces of smooth compctly supported nd rpidly decresing test functions. In generl [61], we cll topologicl vector spce A spce of test functions if D A E, where the inclusions d re continuous nd dense, nd if is continuous opertor on A. An dx useful spce, prticulrly in the study of distributionl symptotic expnsions [21, 22, 36, 56] is K (R), the dul of K(R). The test function

spce K(R) is given by K(R) = α R K α(r), the union hving topologicl mening, where ech K α (R) consists of those smooth functions φ tht stisfy (2.1) φ (m) (t) = O( t α m ) s t, m N, nd is provided with the topology generted by the fmily of seminorms (2.2) mx{sup φ (m) (t), sup t m α φ (m) (t) }. t 1 t 1 The spce K (R) plys fundmentl role in the theory of summbility of distributionl evlutions [13]. We shll use the nottion f, g, F, etc. to denote distributions, while f, g, F, etc. will denote functions. If f is loclly Lebesgue integrble function nd f is the corresponding regulr distribution, given by f, φ = f (x) φ (x) dx for φ D(R), then we shll use the nottion f f; nturlly f is not relly function but n equivlence clss of functions equl lmost everywhere. 2.2. Point vlues. In [30, 31] Lojsiewicz defined the vlue of distribution f D (R) t the point x 0 s the limit (2.3) f(x 0 ) = lim f(x 0 + εx), ε 0 if the limit exists in D (R), tht is if (2.4) lim ε 0 f(x 0 + εx), φ(x) = f(x 0 ) φ(x) dx, for ech φ D(R). It ws shown by Lojsiewicz tht the existence of the distributionl point vlue f(x 0 ) = γ is equivlent to the existence of n N, nd primitive of order n of f, tht is F (n) = f, which corresponds, ner x 0, to continuous function F tht stisfies n!f (x) (2.5) lim x x0 (x x 0 ) n = γ. One cn lso define point vlues by using the opertor (2.6) x0 (f) = ((x x 0 ) f (x)), since f 1 (x 0 ) = γ if nd only if f(x 0 ) = γ, where f = x0 (f 1 ). Therefore [7] f hs distributionl vlue equl to γ t x = x 0 if nd only if there exists n N nd function f n, continuous t x = x 0, with f n (x 0 ) = γ, such tht f = n x 0 (f n ), ner x 0, where f n f n. Suppose tht f S (R) hs the Lojsiewicz point vlue f(x 0 ) = γ. Initilly, (2.4) is only supposed to hold for φ D(R); however, it is shown in [15, 54] tht (2.4) will remin true for ll φ S(R). Actully using the notion of the Cesàro behvior of distribution t infinity 7

8 RICARDO ESTRADA AND JASSON VINDAS [13] explined below, (2.4) will hold [15, 46, 51, 52] if f (x) = O( x β ) (C), s x, φ (x) = O( x α ), strongly x, nd α < 1, α + β < 1. An symptotic estimte is strong if it remins vlid fter differentition of ny order, nmely, (2.1) is stisfied. The notion of distributionl point vlue introduced by Lojsiewicz hs been shown to be of fundmentl importnce in nlysis [7, 12, 33, 35, 48, 49, 57, 59, 60]. It seems to hve been originted in the ide of generlized differentils studied by Denjoy in [9]. There re other notions of distributionl point vlues s tht of Cmpos Ferreir [7, 8], who lso introduced the very useful concept of bounded distributions (see lso [62]). A distribution f is sid to be distributionlly bounded t x 0 if f(x 0 + εx) = O(1) s ε 0 in D (R), i.e., for ech test function f(x 0 + εx), φ(x) = O(1). Distributionl boundedness dmits chrcteriztion [47] similr to tht of Lojsiewicz point vlues, but this time one replces (2.5) by F (x) = O( x x 0 n ). Notice tht the distributionl limit [30] lim x x0 f(x) cn be defined for distributions f D (R \ {x 0 }). If the point vlue f(x 0 ) exists distributionlly then the distributionl limit lim x x0 f(x) exists nd equls f(x 0 ). On the other hnd, if lim x x0 f(x) = L distributionlly then there exist constnts 0,..., n such tht f(x) = f 0 (x) + n j=0 jδ (j) (x x 0 ), where the distributionl point vlue f 0 (x 0 ) exists nd equls L. We my lso consider lterl limits. We sy tht the distributionl lterl vlue f(x + 0 ) exists if f(x + 0 ) = lim ε 0 + f(x 0 +εx) in D (0, ), tht is, (2.7) lim ε 0 + f(x 0 + εx), φ(x) = f(x + 0 ) 0 φ(x) dx, φ D(0, ). Similr definitions pply to f(x 0 ). Notice tht the distributionl limit lim x x0 f(x) exists if nd only if the distributionl lterl limits f(x 0 ) nd f(x + 0 ) exist nd coincide. 2.3. The Cesàro behvior of distributions t infinity. The Cesàro behvior [13, 22] of distribution t infinity is studied by using the order symbols O (x α ) nd o (x α ) in the Cesàro sense. If f D (R) nd α R\ { 1, 2, 3,...}, we sy tht f(x) = O (x α ) s x in the Cesàro sense nd write (2.8) f(x) = O (x α ) (C), s x, if there exists N N such tht every primitive F of order N, i.e., F (N) = f, corresponds for lrge rguments to loclly integrble function, F F, tht stisfies the ordinry order reltion (2.9) F (x) = p(x) + O ( x α+n), s x,

for suitble polynomil p of degree t most N 1. Note tht if α > 1, then the polynomil p is irrelevnt. If we wnt to specify the vlue N, we write (C, N) insted of just (C). A similr definition pplies to the little o symbol. The definitions when x re cler. The elements of S (R) cn be chrcterized by their Cesàro behvior t ±, in fct, f S (R) if nd only if there exists α R such tht f(x) = O (x α ) (C), s x, nd f(x) = O ( x α ) (C), s x. On the other hnd, this is true for ll α R if nd only if f K (R). Using these ides, one cn define the limit of distribution t in the Cesàro sense. We sy tht f D (R) hs limit L t infinity in the Cesàro sense nd write (2.10) lim x f(x) = L (C), if f(x) = L + o(1) (C), s x. The Cesàro behvior of distribution f t infinity is relted to the prmetric behvior of f(λx) s λ. In fct, one cn show [22, 45, 47] tht if α > 1, then f(x) = O (x α ) (C) s x nd f(x) = O ( x α ) (C) s x if nd only if (2.11) f(λx) = O (λ α ) s λ, where the lst reltion holds wekly in D (R), i.e., for ll φ D(R) fixed, f(λx), φ(x) = O (λ α ), λ. A distribution f belongs to the spce K (R) if nd only if it stisfies the moment symptotic expnsion [21, 22], (2.12) f(λx) n=0 ( 1) n µ n δ (n) (x) n!λ n+1, s λ, where the µ n = f (x), x n re the moments of f. 2.4. Evlutions. Let f D (R) with support bounded on the left. If φ E (R) then the evlution f (x), φ (x) will not be defined, in generl. We sy tht the evlution exists in the Cesàro sense nd equls L, written s (2.13) f (x), φ (x) = L (C), if g (x) = L + o (1) (C) s x, where g is the primitive of fφ with support bounded on the left. A similr definition pplies if supp f is bounded on the right. Observe tht if f corresponds to loclly integrble function f with supp f [, ) then (2.13) mens tht (2.14) f (x) φ (x) dx = L (C). 9

10 RICARDO ESTRADA AND JASSON VINDAS Nturlly, this will hold for ny integrtion method we use. If f (x) = n=0 nδ (x n) then (2.13) tells us tht (2.15) n φ (n) = L (C). n=0 In the generl cse when the support of f extends to both nd +, there re vrious different but relted notions of evlutions in the Cesàro sense (or in ny other summbility sense, in fct). If f dmits representtion of the form f = f 1 + f 2, with supp f 1 bounded on the left nd supp f 2 bounded on the right, such tht f j (x), φ (x) = L j (C) exist, then we sy tht the (C) evlution f (x), φ (x) (C) exists nd equls L = L 1 + L 2. This is clerly independent of the decomposition. The nottion (2.13) is used in this sitution s well. It hppens mny times tht f (x), φ (x) (C) does not exist, but the symmetric limit, lim x {g (x) g ( x)} = L, where g is ny primitive of fφ, exists in the (C) sense. Then we sy tht the evlution f (x), φ (x) exists in the principl vlue Cesàro sense [22, 53], nd write (2.16) p.v. f (x), φ (x) = L (C). Observe tht p.v. n= nφ (n) = L (C) if nd only if N L (C) s N while p.v. if A f (x) φ (x) dx L (C) s A. A N nφ (n) f (x) φ (x) dx = L (C) if nd only A very useful intermedite notion is the following [48, 49, 53]. there exists k such tht (2.17) lim x {g (x) g ( x)} = L (C, k), > 0, we sy tht the distributionl evlution exists in the e.v. Cesàro sense nd write (2.18) e.v. f (x), φ (x) = L (C, k), or just e.v. f (x), φ (x) = L (C) if there is no need to cll the ttention to the vlue of k. 2.5. Lojsiewicz distributions. There is clss of distributions tht correspond to ordinry functions, the clss of Lojsiewicz distributions. In generl Lojsiewicz distributions re not regulr distributions, tht is, they correspond to ordinry functions tht re not loclly Lebesgue integrble functions. The simplest clss of distributions tht correspond to functions re those tht come from continuous functions. If f f nd f is continuous then it is n ordinry function: We cn lwys sy wht f (x 0 ) is If

for ny x 0. The function f is not just defined lmost everywhere but it is ctully defined everywhere. Definition 2.1. A distribution f is Lojsiewicz distribution if the distributionl point vlue f (x 0 ) exists for every x 0 R. Definition 2.2. A function f defined in R is clled Lojsiewicz function if there exists Lojsiewicz distribution f such tht (2.19) f (x) = f (x) x R. The correspondence f f is clerly nd uniquely defined in the cse of Lojsiewicz functions nd distributions [30]. The Lojsiewicz functions cn be considered s distributionl generliztion of continuous functions. They re defined t ll points, nd furthermore the vlue t ech given point is not rbitrry but the (distributionl) limit of the function s one pproches the given point. The Lojsiewicz functions nd distributions were introduced in [30]. If f is Lojsiewicz distribution, nd F is primitive, F = f, then F is lso Lojsiewicz distribution. If f is Lojsiewicz distribution nd ψ is smooth function, then ψf is Lojsiewicz distribution nd (2.20) (ψf) (x) = ψ (x) f (x). If f is Lojsiewicz function, f f, then we cn define its definite integrl [2, 30] s (2.21) f (x) dx = F (b) F (), where F = f. The evlution of f on test function φ, f, φ, cn ctully be given s n integrl, nmely, (2.22) f, φ = = f (x) φ (x) dx f (x) φ (x) dx, φ D (R), where supp φ [, b]. We will give rther constructive procedure below (Sections 4 nd 6) to clculte (2.21). If f 0 is Lojsiewicz function, f 0 f 0, defined for x <, nd f 1 is Lojsiewicz function, f 1 f 1, defined for x >, nd if the distributionl lterl limits f 0 ( 0) nd f 1 ( + 0) exist nd coincide, then there is Lojsiewicz function f whose restriction to (, ) is f 0 nd whose restriction to (, ) is f 1. 11

12 RICARDO ESTRADA AND JASSON VINDAS A typicl exmple of Lojsiewicz function is { x α sin x β, x 0, (2.23) s α,β (x) = 0, x = 0, for α C nd β > 0. If H is the Heviside function, then the functions H (±x) s α,β (x) nd their liner combintions re lso Lojsiewicz functions. It is not hrd to see tht this implies tht derivtives of rbitrry order of s α,β, where s α,β s α,β, re lso Lojsiewicz distributions. These re rpidly oscillting functions. However, not ll fst oscillting functions re Lojsiewicz functions. Curiously, the regulr distribution sin (ln x ) is not Lojsiewicz distribution since the distributionl vlue t x = 0 does not exist in the Lojsiewicz sense, even though it exists nd equls 0 in the Cmpos Ferreir sense [7]. 2.6. Distributionlly regulted functions. Another cse when distribution corresponds to function is the cse of regulted distributions, introduced nd studied in [48]. They re generliztions of the ordinry regulted functions [10], which re functions whose lterl limits exist t ll points, lthough they my be different. They re relted to the recently introduced thick points [20]. Definition 2.3. A distribution f is clled regulted distribution if the distributionl lterl limits (2.24) f ( ) x + 0 nd f ( ) x 0, exist x 0 R, nd there re no delt functions t ny point. The sttement tht there re no delt functions t ny point explicitly mens tht for ech φ D(R) nd ny x 0 R (2.25) lim f(x 0 + εx), φ(x) = f(x ε 0 + 0 ) 0 φ(x)dx + f(x + 0 ) 0 φ(x)dx. The reltion (2.25) is known s (pointwise) distributionl jump behvior nd hs interesting pplictions in the theory of Fourier series [23, 50, 53]. If f ( ) ( ) x + 0 = f x 0 then f (x0 ) exists, since these distributions do not hve delt functions, nd therefore we cn define the function (2.26) f (x 0 ) = f (x 0 ), for these x 0. Then f is clled distributionlly regulted function. The function f is defined in the set R \ S, where S is the set of points x 0

where f ( ) ( ) x + 0 f x 0. The set S hs mesure zero since in fct it is countble t the most [48]. One cn ctully define ) ( ) + f x 0 (2.27) f (x 0 ) = f ( x + 0 nd this is defined everywhere. The bsic properties of the distributionlly regulted functions nd the corresponding regulted distributions re the following. If f is regulted distribution, nd F is primitive, F = f, then F is Lojsiewicz distribution. If f is regulted distribution nd ψ is smooth function, then ψf is regulted distribution too. If f is regulted function, f f, then we cn define its definite integrl s (2.28) where F = f. Then (2.29) f, φ = f (x) dx = F (b) F (), 2 f (x) φ (x) dx, φ D (R). As in the cse of Lojsiewicz functions, the integrl tht we will define in Section 4 coincides with (2.28) for distributionlly regulted functions (Theorem 6.6). 2.7. Romnovski s lemm. We shll use the following useful result [38], the Romnovski s lemm, in some of our proofs. See [26] for mny interesting pplictions of this result, nd [24] for generliztions to severl vribles. Theorem 2.4. (Romnovski s lemm) Let F be fmily of open intervls in (, b) with the following four properties: I. If (α, β) F nd (β, γ) F, then (α, γ) F. II. If (α, β) F nd (γ, δ) (α, β) then (γ, δ) F. III. If (α, β) F for ll [α, β] (c, d) then (c, d) F. IV. If ll the intervls contiguous to perfect closed set K [, b] belong to F then there exists n intervl I F with I K. Then (, b) F. Observe tht if we tke K = [, b] in IV we obtin tht F { }, but it my be esier to show this seprtely. 2.8. Mesures. We shll use the following nomenclture. A (Rdon) mesure would men positive functionl on the spce of compctly, 13

14 RICARDO ESTRADA AND JASSON VINDAS supported continuous functions, which would be denoted by integrl nottion such s dµ, or by distributionl nottion, f = f µ, so tht (2.30) f, φ = φ (x) dµ(x), R nd f, φ 0 if φ 0. A signed mesure is rel bounded functionl on the spce of compctly supported continuous functions, denoted s, sy dν, or s g = g ν. Observe tht ny signed mesure cn be written s ν = ν + ν, where ν ± re mesures concentrted on disjoint sets. We shll lso use the Lebesgue decomposition, ccording to which ny signed mesure ν cn be written s ν = ν bs + ν sig, where ν bs is bsolutely continuous with respect to the Lebesgue mesure, so tht it corresponds to regulr distribution, while ν sig is signed mesure concentrted on set of Lebesgue mesure zero. We shll lso need to consider the mesures (ν sig ) ± = (ν ± ) sig, the positive nd negtive singulr prts of ν. 3. The φ trnsform A very importnt tool in our definition of generl distributionl integrl is the φ trnsform. The φ trnsform [11, 37, 48, 51] in one vrible is defined s follows. Let φ D (R) be fixed normlized test function, tht is, one tht stisfies (3.1) φ (x) dx = 1. If f D (R) we introduce the function of 2 vribles F = F φ {f} by the formul (3.2) F (x, t) = f (x + ty), φ (y), where (x, t) H, the hlf plne R (0, ). Nturlly the evlution in (3.2) is with respect to the vrible y. We cll F the φ trnsform of f. Whenever we consider φ trnsforms we ssume tht φ stisfies (3.1). The φ trnsform converges to the distribution s t 0 + [51, 52]: If φ D (R) nd f D (R), then (3.3) lim F (x, t) = f (x), t 0 + distributionlly in the spce D (R), tht is, if ρ D (R) then (3.4) lim t 0 + F (x, t), ρ (x) = f (x), ρ (x). The definition of the φ trnsform tells us tht if the distributionl point vlue [30] f (x 0 ) exists nd equls γ then F (x 0, t) γ s t 0 +,

but ctully F (x, t) γ s (x, t) (x 0, 0) in n ngulr or nontngentil fshion, tht is if x x 0 Mt for some M > 0 (just replce φ(y) by the compct set {φ (y + τ) : τ M}). The ngulr behvior of the φ trnsform t point (x 0, 0) gives us importnt informtion [11, 37, 51] bout the nture of the distribution t x = x 0, even if the ngulr limit does not exist. If x 0 R we shll denote by C x0,θ the cone in H strting t x 0 of ngle θ, (3.5) C x0,θ = {(x, t) H : x x 0 (tn θ)t}. If f D (R) nd x 0 R then we consider the upper nd lower ngulr vlues of its φ trnsform, (3.6) f + φ,θ (x 0) = lim sup (x,t) (x 0,0) (x,t) C x0,θ (3.7) f φ,θ (x 0) = lim inf (x,t) (x 0,0) (x,t) C x0,θ F (x, t), F (x, t). The quntities f ± φ,θ (x 0) re well defined t ll points x 0, but, of course, they could be infinite. For θ = 0, we obtin the upper nd lower rdil limits of the φ trnsform. The following simple result would be useful. Lemm 3.1. Let f D (R) nd x 0 R. If (3.8) f + φ,0 (x 0) = f φ,0 (x 0) = γ, for ll normlized positive test functions φ D(R), then the distributionl point vlue f (x 0 ) exists nd equls γ. Proof. Indeed, (3.8) yields tht lim ε 0 f (x 0 + εx), φ (x) exists nd equls γ for ny positive normlized test function. If we multiply by constnt, we obtin tht the limit exists nd equls γ φ (x) dx for ny positive test function. The result now follows becuse ny test function is the difference of two positive test functions. Indeed, given n rbitrry test function φ D(R), let M = mx x R φ(x). Find positive ϕ D(R) so tht ϕ(x) = 1 for x supp φ, then φ 1 = Mϕ nd φ 2 = φ + φ 1 re positive test functions with φ = φ 2 φ 1. We shll need severl chrcteriztions of positive mesures in terms of the extreme vlues f ± φ,θ (x) of distribution f. The following result ws proved in [51]. 15

16 RICARDO ESTRADA AND JASSON VINDAS Theorem 3.2. Let f D (R). Let U be n open set. Then f is mesure in U if nd only if its φ trnsform F = F φ {f} with respect to given normlized, positive test function φ D (R) stisfies (3.9) f φ,θ (x) 0 x U, for ll ngles θ. Moreover, if the support of φ is contined in [ R, R] nd if (3.9) holds for single vlue of θ > rctn R, then f is mesure in U. We should lso point out tht if there exists constnt M > 0 such tht f φ,θ (x) M, x U, where θ > rctn R, then f is signed mesure in U, whose singulr prt is positive [51]. It is esy to see tht these results re not true if we use rdil limits insted of ngulr ones. An exmple is provided by tking f (x) = δ (x) nd φ D (R) with φ (0) > 0. Actully this exmple shows tht if (3.9) holds for vlue of θ < rctn R, then f might not be mesure. Using the Romnovski s lemm we were ble to prove the ensuing stronger result in [24]. Theorem 3.3. Let f D (R). Let U be n open set. Suppose its φ trnsform F = F φ {f} with respect to given normlized, positive test function φ D (R) with supp φ [ R, R] stisfies (3.10) f + φ,0 (x) 0 lmost everywhere in U, while for ech x U there is constnt M x > 0 such tht (3.11) f φ,θ (x) M x, where θ > rctn R. Then f is mesure in U. Furthermore, one needs the inequlity (3.11) to be true t ll points of U, s the exmple f (x) = δ (x ), where U, shows. However, in our construction of the generl distributionl integrl we shll need to consider the cse when f φ,θ (x) = for x E where E is smll set in the sense tht E ℵ 0. We hve corresponding result in this cse if we sk tht ny primitive of f be Lojsiewicz distribution. Theorem 3.4. Let f D (R). Suppose tht f = F, where F is Lojsiewicz distribution. Let U be n open set. Suppose the φ trnsform F = F φ {f} with respect to given normlized, positive test function φ D (R) with supp φ [ R, R] stisfies (3.12) f + φ,0 (x) 0 lmost everywhere in U,

while there exist countble set E such tht for ech x U \ E there is constnt M x > 0 such tht (3.13) f φ,θ (x) M x, x U \ E, where θ > rctn R. Then f is mesure in U. Proof. Suppose tht U is n open intervl. Let U be the fmily of open subintervls V of U such tht the restriction f V is mesure. We shll use the Theorem 2.4 to prove tht U U. Let us first show tht U { }. Suppose tht E {x n : 1 n < }. Let t 0 1 be fixed nd put (3.14) g n (x) = min { F (y, t) : y x (tn θ)t, n 1 t t 0 }. The functions g n re continuous nd becuse of (3.13), for ech x U \ E there exists constnt M x > 0 such tht g n (x) M x, for ll n. Hence if (3.15) W k = {x U : g n (x) k n N} {x 1,..., x k }, then U = k=1 W k. If we now employ the Bire theorem we obtin the existence of k N, such tht W k hs non-empty interior, nd thus the interior of the set (3.16) {x U : g n (x) k n N} is lso non-empty. Hence there is non-empty open intervl V U nd constnt M > 0 such tht F (x, t) M for ll (y, t) C x,θ with x V nd 0 < t t 0, nd hence f φ,θ (x) M for x V. The Theorem 3.3 then yields tht f V is mesure. Therefore V U, nd so U { }. Condition I of the Theorem 2.4 follows from the fct tht if f (α,β) nd f (β,γ) re mesures, then F (α,β) nd F (β,γ) re distributions corresponding to incresing continuous functions, nd since F is Lojsiewicz distribution it follows tht F, F F, must lso be continuous t x = β, so tht F is continuous incresing function in (α, γ) nd consequently f (α,γ) is mesure. It is cler tht II nd III re stisfied. In order to prove IV, let K U be perfect closed set such tht ll the intervls contiguous to K belong to U. Then using the Bire theorem gin, there exists n open intervl V U nd constnt M > 0 such tht f φ,θ (x) M for ll x K V. But f is mesure in V \ K, nd thus f φ,θ (x) 0 for x V \ K. The Theorem 3.3 llows us to conclude tht f V is mesure, nd thus V U; this proves IV. 17

18 RICARDO ESTRADA AND JASSON VINDAS Observe tht if the hypotheses of the Theorem 3.4 re stisfied then f is mesure in U, nd thus f φ,θ (x) 0 t ll points of U nd for ll ngles, not just rdilly lmost everywhere, nd similrly the set E where f φ,θ (x) = is ctully empty. We shll lso employ chrcteriztions merely in terms of rdil limits of the φ trnsform. The following is such result for the lower rdil limits of hrmonic function. Theorem 3.5. Let H (x, y) be hrmonic function defined in the upper hlf plne H. Suppose tht lim (x,y) H (x, y) = 0. Also suppose tht the distributionl limit of H (x, y) s y 0 + exists nd equls f E (R) ; suppose tht f = F, where F is Lojsiewicz distribution. If (3.17) lim sup y 0 + H (x, y) 0 lmost everywhere in R, nd there exists countble set E nd constnts M x < for x R\E such tht (3.18) lim inf y 0 + H (x, y) M x, x R \ E, then f is mesure nd H (x, y) 0 for ll (x, y) H. Proof. We shll employ Romnovski s lemm, Theorem 2.4 to prove tht f is mesure in R. Let (, b) be n open intervl with supp f (, b). Let U be the fmily of open subintervls of (, b) where the restriction of f is mesure; clerly U contins non empty intervls. Observe tht if (c, d) U, then F is n incresing continuous function in [c, d], where F F; condition I follows from this observtion. Conditions II nd III re esy. For condition IV, suppose tht K is perfect compct subset of (, b) such tht (, b) \ K = n=1 ( n, b n ), with ( n, b n ) U. Let m = min x R H (x, 1). By the Bire theorem, there exists constnt M, with M > 0 nd M > m, nd n open intervl I, such tht I K nd H (x, y) M for x I K nd for 0 < y 1. If ( n, b n ) I, then the hrmonic function H is bounded below by M in the boundry of the rectngle ( n, b n ) (0, 1) H, except perhps t the corners n nd b n, but since f is the derivtive ( of Lojsiewicz ((x distribution we obtin the bound H (x, y) = o x0 ) 2 + y 2) ) 1/2 s (x, y) x 0, for ny x 0 R, nd this llows to conclude tht H is bounded below by M in the rectngle [ n, b n ] (0, 1]. Actully if H were not bounded below in the rectngle then t one of the corners, x 0 = n or x 0 = b n, H would grow s fst s or fster thn ( (x x 0 ) 2 + y 2) 1, s follows from the results of [15, Section 4] when pplied to the hrmonic function H(ξ) = H( ξ x 0 ). Therefore

H (x, y) M for ll x I nd ll 0 < y 1, nd the fct tht I U is obtined. It is convenient to define some clsses of test functions. Definition 3.6. The clss T 0 consists of ll positive normlized functions φ E (R) tht stisfy the following condition: (3.19) α < 1 such tht φ (x) = O ( x α ) strongly s x. The clss T 1 is the subclss of T 0 consisting of those functions tht lso stisfy (3.20) xφ (x) 0 for ll x R. 19 Observe tht the φ trnsform is well defined when f E (R) nd φ T 0. Since the Poisson kernel ϕ (x) = π 1 (1 + x 2 ) 1 belongs to T 1 nd the φ trnsform H = F ϕ {f} with respect to this function ϕ is the hrmonic function H (x, y) defined for (x, y) H, tht vnishes t infinity, nd tht stisfies H (x, 0 + ) = f (x) distributionlly, we then hve the following result, corollry of the Theorem 3.5. Theorem 3.7. Let f E (R). Suppose tht f = F, where F is Lojsiewicz distribution. Suppose tht the φ trnsform F = F φ {f} with respect ny φ T 1 stisfies (3.21) f + φ,0 (x) 0 lmost everywhere in R, (3.22) f φ,0 (x) M x >, x R \ E, where E is countble set. Then f is mesure in R. 4. The definite integrl Let f be function defined in [, b] with vlues in R = R {, }. We now proceed to define its integrl. We strt with the concepts of mjor nd minor pirs. Definition 4.1. A pir (u, U) is clled mjor distributionl pir for the function f if: 1) u E [, b], U D (R), nd (4.1) U = u. 2) U is Lojsiewicz distribution, with U () = 0.

20 RICARDO ESTRADA AND JASSON VINDAS 3) There exists set E, with E ℵ 0, nd set of null Lebesgue mesure Z, m (Z) = 0, such tht for ll x [, b] \ Z nd ll φ T 0 we hve (4.2) (u) φ,0 (x) f (x), while for x [, b] \ E nd ll φ T 1 (4.3) (u) φ,0 (x) >. The definition of minor distributionl pir is similr. Definition 4.2. A pir (v, V) is clled minor distributionl pir for the function f if: 1) v E [, b], V D (R), nd (4.4) V = v. 2) V is Lojsiewicz distribution, with V () = 0. 3) There exists set E, with E ℵ 0, nd set of null Lebesgue mesure Z, m (Z) = 0, such tht for ll x [, b] \ Z nd ll φ T 0 we hve (4.5) (v) + φ,0 (x) f (x), while for x [, b] \ E nd ll φ T 1 (4.6) (v) + φ,0 (x) <. Nturlly, we my lwys ssume in the Definitions 4.1 nd 4.2 tht the countble set stisfies E Z. Employing the results of the Theorem 3.7, we immeditely obtin the following useful result. Lemm 4.3. If (u, U) is mjor distributionl pir nd (v, V) is minor distributionl pir for f, then u v is positive mesure nd U V is continuous incresing function, where U U nd V V. If (u, U) is mjor distributionl pir nd (v, V) is minor distributionl pir for f, then U nd V re constnt in the intervl [b, ), nd V (b) U (b). Definition 4.4. A function f : [, b] R is clled distributionlly integrble if it hs both mjor nd minor distributionl pirs nd if (4.7) sup V (b) = inf U (b). (v,v) minor pir (u,u) mjor pir

When this is the cse this common vlue is the integrl of f over [, b] nd is denoted s (4.8) (dist) f (x) dx, or just s f (x) dx if there is no risk of confusion. We shll show in Section 6 tht ny Lebesgue integrble function, nd more generlly, ny Denjoy-Perron-Henstock integrble function is distributionlly integrble, nd the integrls re the sme. Therefore the symbol f (x) dx will hve only one possible mening if the function f is Denjoy-Perron-Henstock integrble or Lebesgue integrble. In some cses we shll use the nottion (dist) f (x) dx, however, to emphsize tht we re deling with the integrl defined in this rticle. Oc- csionlly, we shll lso use the nottion (D P H) f (x) dx for Denjoy-Perron-Henstock integrl nd (Leb) f (x) dx for Lebesgue integrl. Observe tht the function f is distributionlly integrble over [, b] if nd only if for ech ε > 0 there re minor nd mjor pirs, (v, V) nd (u, U), such tht (4.9) U (b) V (b) < ε. We shll first show tht the distributionl integrl hs the stndrd properties of n integrl. Proposition 4.5. If f is distributionlly integrble over [, b] then it is distributionlly integrble over ny subintervl [c, d] [, b]. Proof. Let ε > 0, nd choose minor nd mjor pirs for f over [, b], (v, V) nd (u, U), such tht U (b) V (b) < ε. Let U U nd V V. Let now Ũ nd Ṽ be the Lojsiewicz distributions corresponding to the Lojsiewicz functions Ũ nd Ṽ given by 0 x < c, (4.10) Ũ (x) = U (x) U (c), c x d, U (d) U (c), x > d, nd 0 x < c, (4.11) Ṽ (x) = V (x) V (c), c x d, V (d) V (c), x > d. (Ṽ Ṽ) (Ũ Ũ) Then, nd, re minor nd mjor pirs for f over [c, d], nd Ũ (d) Ṽ (d) < ε. 21

22 RICARDO ESTRADA AND JASSON VINDAS We now consider the integrls of functions tht re equl lmost everywhere. As it is the cse with other integrls, the integrl cn ctully be defined s functionl on the spce of equivlence clsses of functions equl (.e.), nd ech clss hs elements tht re finite everywhere. Proposition 4.6. If f is distributionlly integrble over [, b] then it is finite lmost everywhere. Proof. Let A be the set of points where f (x) =. Let (v, V) nd (u, U) be minor nd mjor pirs for f over [, b], nd let E be the denumerble set outside of where (u) φ,0 (x) > nd (v)+ φ,0 (x) <, for ll φ T 1. Consider the incresing continuous function ρ (x) = U (x) V (x). Using (4.3) nd (4.6) we obtin tht if x A \ E then ρ (x) =, but the set of points where the derivtive of n incresing continuous function is infinite hs mesure 0. The ensuing result llows us to consider distributionl integrtion of functions tht re defined lmost everywhere. Proposition 4.7. If f is distributionlly integrble over [, b] nd g (x) = f (x) (.e.) then g is lso distributionlly integrble over [, b] nd (4.12) g (x) dx = f (x) dx. Proof. Indeed, ny mjor or minor pir for f is lso mjor or minor pir for g, nd conversely. The integrl hs the expected liner properties. Proposition 4.8. If f 1 nd f 2 re distributionlly integrble over [, b] then so is f 1 + f 2 nd (4.13) (f 1 (x) + f 2 (x)) dx = f 1 (x) dx + f 2 (x) dx. Proof. Using Propositions 4.6 nd 4.7 it follows tht we my ssume tht both f 1 nd f 2 re finite everywhere, so tht its sum is lso defined everywhere. Then we just observe tht the sum of mjor pirs for f 1 nd f 2 is mjor pir for f 1 + f 2, nd similrly for the sum of minor pirs.

Proposition 4.9. If f is distributionlly integrble over [, b] then so is kf for ny constnt k nd (4.14) kf (x) dx = k f (x) dx. Proof. The result follows from the following observtions. If k > 0 then multiplying mjor pir for f with k gives mjor pir for kf, nd similrly for minor pirs. If k < 0 then multipliction with k trnsforms mjor pirs for f into minor pirs for kf nd minor pirs for f into mjor pirs for kf. It follows from the previous results tht the set of distributionlly integrble functions over [, b] is liner spce nd tht the integrl is liner functionl. We lso hve the following esy result. Proposition 4.10. Suppose < c < b. A function f defined in [, b] is distributionlly integrble there if nd only if it is distributionlly integrble over [, c] nd [c, b], nd when this is the cse, (4.15) f (x) dx = c f (x) dx + c f (x) dx. 23 If A [, b] then we sy tht f is distributionlly integrble over A if χ A f, where χ A is the chrcteristic function of A, is distributionlly integrble, nd use the nottion (4.16) (dist) f (x) dx. A As with ny non-bsolute integrl, f will not be integrble over ll mesurble subsets of [, b], but if A hs mesure 0 the distributionl integrl exists nd equls 0. Also, ccording to Proposition 4.5, if f is distributionlly integrble over [, b] then it is integrble over ny of its subintervls. 5. The indefinite integrl We shll now study the indefinite integrl function (5.1) F (x) = (dist) x f (t) dt, of function f tht is distributionlly integrble over [, b]. We re interested in the cse when x b, but sometimes it would be

24 RICARDO ESTRADA AND JASSON VINDAS convenient to extend the domin of F by putting F (x) = 0 for x < nd F (x) = F (b) for x > b. The indefinite integrl of Lebesgue integrble function is bsolutely continuous, while tht of Denjoy-Perron-Henstock integrble function is continuous. We shll show tht (5.1) defines Lojsiewicz function, with ssocited Lojsiewicz distribution F, F F. We shll lso show tht the derivtive f = F is distribution tht hs Lojsiewicz distributionl point vlues lmost everywhere nd ctully f (x) = f (x) (.e.). We strt with some useful results. Lemm 5.1. Let (v, V) nd (u, U) be minor nd mjor pirs for distributionlly integrble function f over [, b]. Let U U nd V V. Then U F nd F V re both continuous incresing functions tht vnish t x =. Proof. Observe tht if c < d b then (4.10) gives mjor pir (Ũ Ũ), for f over [c, d] with Ũ (t) = U (t) U (c) for c x d. Thus nd so F (d) F (c) = d c f (x) dx Ũ (d) = U (d) U (c), (5.2) U (c) F (c) U (d) F (d). Similrly one shows tht F V is incresing. Observe now tht U V = (U F ) + (F V ) is continuous incresing function (Lemm 4.3) written s the sum of two incresing functions: we conclude tht both U F nd F V re continuous. Using the lemm we see tht F = (F V )+V is the sum of continuous function nd Lojsiewicz function nd thus it is Lojsiewicz function. Theorem 5.2. Let f be distributionlly integrble function over [, b], with indefinite integrl F. Then F is Lojsiewicz function. Observe tht one my consider f s n equivlence clss of functions defined lmost everywhere, nd thus the vlue f (x) for prticulr x my or my not hve useful mening. However, F is Lojsiewicz function, nd this implies tht the vlue F (x) hs cler interprettion for ll numbers x. Since F is Lojsiewicz function, it hs n ssocited Lojsiewicz distribution F. The distributionl derivtive f = F is well defined

distribution with supp f [, b]. The reltionship between f nd f is s follows. Theorem 5.3. Let f be distributionlly integrble function over [, b], with indefinite integrl F, let F F, nd let f = F. Then f hs point vlues lmost everywhere nd (5.3) f (x) = f (x) (.e.). Proof. Let ε, η > 0. Let (u, U) be mjor pir for f over [, b] with (5.4) U (b) F (b) < εη, where U U. Let ρ = U F, n incresing continuous function. Consider the set A = {x [, b] : ρ (x) ε}. Since εm (A) ρ (x) dx ρ (b) < εη, it follows tht m (A) < η, where m (A) is the Lebesgue mesure. Notice now tht if x [, b] \ (A Z), where Z is the null set outside of where (u) φ,0 (x) f (x) >, for ll φ T 0, then Hence (5.5) m (f) φ,0 (x) = (u) φ,0 (x) ρ (x) > f (x) ε. ({ }) x [, b] : (f) φ,0 (x) f (x) ε φ T 0 < η. But η is rbitrry, nd thus the set where (f) φ,0 (x) f (x) ε hs mesure 0, nd since ε is lso rbitrry we obtin tht (f) φ,0 (x) f (x) (.e.). Using similr nlysis involving minor pirs one likewise obtins tht (f) + φ,0 (x) f (x) (.e.). If we now use the Lemm 3.1 then (5.3) follows. The following consequence of the preceding theorem is worth mentioning. Corollry 5.4. If f is distributionlly integrble over [, b] then it is mesurble. Proof. Let φ D (R) be normlized test function. Then the sequence of continuous functions (5.6) f n (x) = f (x + y/n), φ (y)), converges to f lmost everywhere, nmely where (5.3) holds, nd the mesurbility of f is thus obtined. 25

26 RICARDO ESTRADA AND JASSON VINDAS If we now use Theorem 5.3, combined with Lemm 5.1, we obtin more informtion on the nture of mjor nd minor pirs. Proposition 5.5. Let (v, V) nd (u, U) be minor nd mjor pirs for distributionlly integrble function f over [, b]. Then the distributionl point vlues v (t) nd u (t) exist lmost everywhere in [, b]. If ṽ is function given by the point vlues of v, nmely, ṽ (t) = v (t) when the vlue exists, extended in ny wy to function over [, b], then ṽ is distributionlly integrble over [, b]. Similrly the function ũ (t) = u (t), when the vlue exists, is distributionlly integrble over [, b]. Furthermore, (5.7) V (d) V (c) nd (5.8) d c f (x) dx d c d c ṽ (x) dx d c f (x) dx, ũ (x) dx U (d) U (c). Proof. Let U U, V V, nd F F. Since F V is n incresing continuous function, it follows tht (F V) = f v is positive mesure, nd thus it hs distributionl vlues lmost everywhere, nd since f hs.e. distributionl vlues (equl to f), it follows tht likewise v hs distributionl vlues.e.. The function ṽ is distributionlly integrble becuse ṽ(t) = f(t) h(t) (.e.), where h is the Lebesgue integrble function which corresponds to the bsolutely continuous prt of f v (see Theorem 6.1 below). The inequlity (5.7) is obtined from the fct tht (5.9) 0 d c (f (x) ṽ (x)) dx (F (d) V (d)) (F (c) V (c)). The results for the mjor pir re obtined in similr fshion. This proposition suggests n lterntive pproch to the distributionl integrl. Cll pir (u, U) mjor pir v.2 (version 2) if it stisfies ll the conditions of the Definition 4.1 plus the extr requirement tht u (x) exist lmost everywhere in [, b]. Define, nlogously, minor pirs v.2 nd n integrl in terms of mjor nd minor pirs v.2. Then this integrl would be identicl to the distributionl integrl we hve been considering, becuse ny mjor or minor pir in the originl sense is ctully pir in the v.2 sense. However, use of the definition v.2 llows one to obtin some proofs, s tht of Theorems 5.2 nd 5.3, in rther simple wy. Propositon 5.5 lso hs the following consequence on the mjor nd minor distributionl pirs.

Corollry 5.6. Let (v, V) nd (u, U) be minor nd mjor pirs for distributionlly integrble function f over [, b]. Then, there exists set of null Lebesgue mesure Z such tht for ll x [, b] \ Z, ll φ T 0, nd ll ngles we hve (5.10) (u) φ,θ (x) f (x), nd (5.11) (v) + φ,θ (x) f (x). Proof. Let Z be the complement in [, b] of the set on which the distributionl point vlues of u, v, nd f exist. Then Z hs null Lebesgue mesure nd (5.10) nd (5.11) re both vlid on [, b] \ Z. Corollry 5.6 implicitly suggests third vrint yet for the definition of the distributionl integrl. Let us sy tht (u, U) is mjor pir v.3 (version 3) if it stisfies the conditions of Definition 4.1 nd dditionlly we replce the rdil condition (4.3) by the stronger requirement (5.10), ssumed to hold for ll x [, b] \ Z, m(z) = 0, ll φ T 0, nd ll ngles. Likewise, one defines minor pirs v.3. If we define n integrl in terms of mjor nd minor pirs v.3, then we obtin nothing new, becuse in view of Corollry 5.6 this integrl coincides with the distributionl integrl defined in Section 4. 27 6. Comprison with other integrls We shll now consider the reltionship of the distributionl integrl to the Lebesgue integrl, to the Denjoy-Perron-Henstock, nd to the Lojsiewicz method (see (2.21)). We lso give constructive solution to Denjoy s problem on the reconstruction of functions from their higher order differentil quotients [9]. Let us strt with the Lebesgue integrtion. Theorem 6.1. Any Lebesgue integrble function over [, b] is lso distributionlly integrble over [, b] nd the integrls coincide. Proof. Let ε > 0. If f is Lebesgue integrble function over [, b], we cn pply the Vitli-Crthéodory Theorem [40, III (7.6)] to find lower semi-continuous function u with u (x) f (x) for ll x, nd with (6.1) (Leb) (u (x) f (x)) dx < ε 2.

28 RICARDO ESTRADA AND JASSON VINDAS If U (x) = x u (x) dx, then the pir (U, U), where U U, is distributionl mjor pir for f; with (6.2) U (b) < (Leb) f (x) dx + ε 2. Similrly, employing minor functions nd upper semi-continuous functions, we cn find minor distributionl mjor pir for f, (V, V) with (6.3) V (b) > (Leb) f (x) dx ε 2. The distributionl integrbility of f nd the fct tht (6.4) (dist) then follow. f (x) dx = (Leb) f (x) dx, The Perron method of integrtion uses mjor nd minor functions [26, 34, 40]. We shll show tht these functions give mjor nd minor distributionl pirs in nturl wy. Theorem 6.2. Any Denjoy-Perron-Henstock integrble function over [, b] is lso distributionlly integrble over [, b] nd the integrls coincide. Proof. Let U be continuous mjor function for Denjoy-Perron- Henstock integrble function f over [, b]. Then the pir (U, U), where U U, is distributionl mjor pir for f. Indeed, the derivtive U (x) exists (.e.) in [, b], nd t those points the distributionl vlue U (x) exists, nd thus (U ) φ,0 (x) = U (x) = U (x) f (x) for ll φ T 0. Furthermore, for ny x [, b], (6.5) lim inf y x U (y) U (x) y x >. But if (y x) 1 (U (y) U (x)) M for x y < c, then we cn write U = U 1 + U 2, where U 1 (y) = χ (x c,x+c) (y) U 1 (y). Let φ T 1. Since U 2 (y) = 0 in neighborhood of y = x, it follows tht U 2 (x + εy), φ (y) 0. Also, 1 lim inf U ε 0 + 1 (x + εy), φ (y) = lim inf ε 0 + ε U 1 (x + εy) U 1 (x), φ (y) M = M. yφ (y) dy