Park, Ryu, and Lee recently introduced a Henstock-type integral, which lies between the Mcshane and the Henstock integrals. This paper proves the closure property of this new integral under Cauchy extension, and presents a characterization on absolute Mα-integrabilit

Garces, Ian June L

Racca, Abraham P

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archīum.ATENEO (Ateneo de Manila Univ.)

Ateneo de Manila University Archīum Ateneo Mathematics Faculty Publications Mathematics Department 2015 Cauchy Extension of Mα-Integral and Absolute Mα-Integrability Ian June L. Garces Abraham P. Racca Follow this and additional works at: https://archium.ateneo.edu/mathematics-faculty-pubs  Part of the Mathematics Commons International Journal of Mathematical AnalysisVol. 9, 2015, no. 27, 1323 - 1329HIKARI Ltd, www.m-hikari.comhttp://dx.doi.org/10.12988/ijma.2015.5368Cauchy Extension of Mα-Integral andAbsolute Mα-IntegrabilityI. J. L. GarcesDepartment of MathematicsAteneo de Manila University1108 Loyola Heights, Quezon CityThe PhilippinesAbraham P. RaccaMathematics and Computer Science DepartmentAdventist University of the Philippines4118 Silang, Cavite, The PhilippinesCopyright c© 2015 I. J. L. Garces and Abraham P. Racca. This article is distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.AbstractPark, Ryu, and Lee recently introduced a Henstock-type integral,which lies between the Mcshane and the Henstock integrals. This paperproves the closure property of this new integral under Cauchy extension,and presents a characterization on absolute Mα-integrability.Mathematics Subject Classification: 26A39Keywords: Mα-integral, Cauchy extension, absolute Mα-integrable1 IntroductionPark, Ryu, and Lee [4] recently defined a Henstock-type integral, which theycall Mα-integral. Several properties and convergence theorems of the integralwere established in [1] and [4]. Most of them paralleled the usual properties1324 I. J. L. Garces and A. P. Raccaof Henstock integral, including the Saks-Henstock Lemma [4, Lemma 2.5].Moreover, by providing examples, it was also shown in [4] that Mα-integrallies strictly between the McShane integral (which is known to be equivalent tothe Lebesgue integral (see [2])) and the Henstock integral.Let α > 0 be a constant, and I = [a, b] a non-degenerate closed andbounded interval in R.(1) A partial division D of I is a finite collection of interval-point pairs([u, v], ξ) such that the closed intervals [u, v] are non-overlapping,⋃[u, v] ⊆I, and ξ ∈ I. If ⋃[u, v] = I, we call the division D simply a division.(2) A positive function defined on I is called a gauge on I.(3) Let δ be a gauge on I, and D = {([u, v], ξ)} a partial division of I. If[u, v] ⊆ (ξ − δ(ξ), ξ + δ(ξ)) for all ([u, v], ξ) ∈ D, then we say that D isa δ-fine McShane partial division. Moreover, if D is a McShane partialdivision such that ξ ∈ [u, v] for all ([u, v], ξ) ∈ D, then D is called aδ-fine Henstock partial division.(4) A McShane division D = {([u, v], ξ)} of I is said to be an Mα-division if∑([u,v],ξ)∈Ddist(ξ, [u, v]) < α,where dist(x, J) = inf{|y − x| : y ∈ J}.(5) Let D = {([u, v], ξ)} be a partial division on I, and f a real-valuedfunction defined on I. We writeS(f,D) =∑([u,v],ξ)∈Df(ξ)(v − u).With these terms and notations, the definition of Mα-integrability can nowbe presented.Definition 1.1 ([4, Definition 2.1]). A function f : [a, b]→ R is Mα-integrableif there exists a real number A such that, for each  > 0, there is a gauge δ on[a, b] such that|S(f,D)− A| < for each δ-fine Mα-division D of [a, b]. Here, A is called the Mα-integral of fon [a, b], and we write A =∫ baf .Moreover, f is Mα-integrable on E ⊆ [a, b] if fχE is Mα-integrable on [a, b],where χE is the characteristic function over E, and we write∫ bafχE =∫Ef .This paper proves the closure property of this new integral under Cauchyextension (Theorem 2.1) and a characterization on absolute Mα-integrability(Theorem 3.3).Cauchy extension of Mα-integral and absolute Mα-integrability 13252 Cauchy ExtensionThe following theorem shows that Mα-integral is closed under Cauchy exten-sion, a property that is not valid for McShane integral.Theorem 2.1 (Cauchy Extension). Let f : [a, b]→ R be Mα-integrable on [c, b]for each c ∈ (a, b). Then f is Mα-integrable on [a, b] if and only if limc→a+∫ bcfexists. In this case, we have ∫ baf = limc→a+∫ bcf.Proof. Suppose that f is Mα-integrable on [a, b], and let  > 0 be given. Thenthere exists a gauge δ on [a, b] such that if D is a δ-fine Mα-division of [a, b],then ∣∣∣∣S(f,D)− ∫ baf∣∣∣∣ < .Moreover, for each c ∈ (a, b), there exists a gauge δc on [c, b] such that if Dc isa δc-fine Mα-division of [c, b], then∣∣∣∣S(f,Dc)− ∫ bcf∣∣∣∣ < .For each c ∈ (a, b), we may assume that δc(x) ≤ δ(x) for all x ∈ [c, b].Choose s ∈ (a, a + δ(a)) such that |f(a)|(s − a) < . Let Ds be a δs-fineMα-division of [s, b]. Then D = Ds ∪ {([a, s], a)} is a δ-fine Mα-division of[a, b]. Hence, we have∣∣∣∣∫ baf −∫ bsf∣∣∣∣ ≤ ∣∣∣∣∫ baf − S(f,D)∣∣∣∣+ ∣∣∣∣S(f,Ds)− ∫ bsf∣∣∣∣+ |f(a)|(s− a) < 3.Since s is chosen arbitrarily close to a, we obtain limc→a+∫ bcf exists and isequal to∫ baf .Conversely, suppose that limc→a+∫ bcf = A exists, and let {x0, x1, x2, . . .}be a strictly decreasing sequence of real numbers such that x0 = b and xn → a.Then f is Mα-integrable on [xn, xn−1] for every n ∈ Z+.Let  > 0 be given. For each integer n ≥ 1, choose a gauge δn on [xn, xn−1]such that if Dn is a δn-fine Mα-division of [xn, xn−1], then∣∣∣∣S(f,Dn)− ∫ xn−1xnf∣∣∣∣ < 2n .Furthermore, choose an integer N > 0 such that∣∣∣∣∫ btf − A∣∣∣∣ <  and |f(a)|(t− a) < 1326 I. J. L. Garces and A. P. Raccafor every t ∈ (a, xN ].Define a gauge δ on [a, b] as follows:δ(x) =xN − x if x = amin{δ1(x), x− x1} if x ∈ (x1, x0]min{δn(x), x− xn, xn−1 − x} if x ∈ (xn, xn−1), n ≥ 2min{δn(x), δn+1(x), xn−1 − x, x− xn+1} if x = xn, n ≥ 1.Let D be a δ-fine Mα-division of [a, b]. ThenD = {([a, t], a)} ∪Dt ∪m⋃i=1Di,where t ∈ (xm+1, xm] for some integer m ≥ N , Dt a δm+1-fine Mα-divisionof [t, xm] ⊂ (xm+1, xm], and Di a δi-fine Mα-division of [xi, xi−1]. Thus, bySaks-Henstock Lemma, we have|S(f,D)− A| ≤ |f(a)|(t− a) +∣∣∣∣S(f,Dt)− ∫ xmtf∣∣∣∣+m∑i=1∣∣∣∣S(f,Di)− ∫ xi−1xif∣∣∣∣+ ∣∣∣∣∫ btf − A∣∣∣∣< +2m+1+∞∑i=12i+ < 4.This ends the proof of the theorem.Corollary 2.2. Let f : [a, b] → R be Mα-integrable on every [c, d] ⊂ (a, b).Then f is Mα-integrable on [a, b] if and only if limc→a+d→b−∫ dcf exists. In thiscase, we have ∫ baf = limc→a+d→b−∫ dcf.3 Absolute Mα-IntegrabilityRecall that an integral theory is absolute if a function f is integrable (in thesense of that theory) if and only if |f | is also integrable. Both Riemann andMcShane integrals are absolute (see [2] or [5]), but Henstock integral is knownto be not (see [3]).Cauchy extension of Mα-integral and absolute Mα-integrability 1327Example 3.1. Let∑∞n=1 an be a convergent series. Define a function f :[0, 1]→ R as follows:f(x) ={2nan if x ∈(12n, 12n−1], n ∈ Z+0 if x = 0.Clearly, f is Mα-integrable on [c, 1] for each c ∈ (0, 1). Thus, by Theorem 2.1,we have ∫ 10f = limc→0+∫ 1cf =∞∑n=1an.However, it is not difficult to see that |f | would be Mα-integrable if and only ifthe given series is absolutely convergent.In general, due to the closure under Cauchy extension, it is possible for afunction f to be Mα-integrable on [a, b], but |f | is not. We have the followingproposition.Proposition 3.2. The Mα-integral is not absolute.Let a function f : [a, b]→ R be Mα-integrable on [a, b]. Then the functionF : [a, b] → R, defined as F (a) = 0 and F (x) = ∫ xaf for x ∈ (a, b], is calledthe primitive of f on [a, b].Recall that a function F : [a, b]→ R is said to be of bounded variation on[a, b] ifV (F, [a, b]) = sup{∑P|F (u, v)|}<∞,where the supremum runs over all partitions P of [a, b] (that is, a finitecollection of non-overlapping intervals [u, v] such that⋃[u, v] = [a, b]), andF (u, v) = F (v)− F (u).The following theorem answers the question on exactly when |f | is Mα-integrable on [a, b], given that f is Mα-integrable on [a, b].Theorem 3.3. Let a function f : [a, b] → R be Mα-integrable on [a, b] withprimitive F . Then |f | is Mα-integrable on [a, b] if and only if F is of boundedvariation over [a, b]. In this case, we have∫ ba|f | = V (F, [a, b]).Proof. Suppose that |f | is Mα-integrable on [a, b]. Let P = {[u, v]} be apartition of [a, b]. Then∑P|F (u, v)| =∑P∣∣∣∣∫ vuf∣∣∣∣ ≤∑P∫ vu|f | =∫ ba|f |,1328 I. J. L. Garces and A. P. Raccawhich implies V (F, [a, b]) <∞. Thus, F is of bounded variation over [a, b].Conversely, suppose that F is of bounded variation over [a, b]. Let  > 0 begiven. Then there exists a partition P = {[xi−1, xi] : i = 1, 2, . . . , n} of [a, b]such thatV (F, [a, b])− 2<n∑i=1|F (xi−1, xi)| ≤ V (F, [a, b]).Moreover, by the Mα-integrability of f , there exists a gauge δ0 on [a, b] suchthat|S(f,D)− F (a, b)| < 4for any δ0-fine Mα-division D of [a, b].Defineδ(x) =min{δ0(x), x− xi−1, xi − x} if x ∈ (xi−1, xi) for some imin{δ0(x), x− xi−1, xi+1 − x} if x = xi, 1 ≤ i ≤ n− 1min{δ0(x), x1 − x} if x = x0 = amin{δ0(x), x− xn−1} if x = xn = b,and let D = {([uk, vk], ξk) : k = 1, 2, . . . ,m} be a δ-fine Mα-division of [a, b].By the definition of δ, D has x0 = a, x1, x2, . . . , xn = b as among its tags.Further, if ([u, v], ξ) ∈ D, where ξ ∈ {x0, x1, . . . , xn}, then ξ ∈ [u, v], so thatthese tags can be made as endpoints andV (F, [a, b])− 2<n∑i=1|F (xi−1, xi)| ≤m∑k=1|F (uk, vk)| ≤ V (F, [a, b]).Thus, applying Saks-Henstock Lemma and triangle inequality, we have∣∣S(|f |, D)− V (F, [a, b])∣∣ ≤ ∣∣∣∣∣m∑k=1(|f(ξk)|(vk − uk)− |F (uk, vk)|)∣∣∣∣∣+∣∣∣∣∣m∑k=1|F (uk, vk)| − V (F, [a, b])∣∣∣∣∣<m∑k=1|f(ξ)(vk − uk)− F (uk, vk)|+ 2< .This ends the proof of the theorem.Corollary 3.4. Let the functions f, g : [a, b] → R be Mα-integrable on [a, b]with |f(x)| ≤ g(x) for all x ∈ [a, b]. Then |f | is Mα-integrable on [a, b], and∣∣∣∣∫ baf∣∣∣∣ ≤ ∫ ba|f | ≤∫ bag.Cauchy extension of Mα-integral and absolute Mα-integrability 1329References[1] I. J. L. Garces and A. P. Racca, Characterizing Convergence Conditionsfor the Mα-Integral, J. Chungcheong Math. Soc. 24 (3) (2011) 469-480.[2] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Per-ron, and Henstock, American Mathematical Society, 1994.http://dx.doi.org/10.1090/gsm/004[3] P. Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific,1989. http://dx.doi.org/10.1142/0845[4] J. M. Park, H. W. Ryu and H. K. Lee, The Mα-Integral, J. ChungcheongMath. Soc. 23 (1) (2010) 99-108.[5] H. L. Royden and P. M. Fitzpatrick, Real Analysis, Pearson, 2010.Received: March 19, 2015; Published: April 30, 2015

Cauchy Extension of Mα-Integral and Absolute Mα-Integrability

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Cauchy Extension of Mα-Integral and Absolute Mα-Integrability

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