EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 2, 2020, 303-313 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global McShane Integrability Using Variational Measure Felipe R. Sumalpong, Jr.1,∗, Julius V. Benitez1 1 Department of Mathematics and Statistics, College of Sciences and Mathematics, Mindanao State University-Iligan Institute of Technology, Tibanga, Iligan City, Philippines Abstract. If f : [a, b]→ R is McShane integrable on [a, b], then f is McShane integrable on every Lebesgue measurable subset of [a, b]. However, integrability of a real-valued function on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. In this paper, we give a characterization for the McShane integrability of f : [a, b]→ R over E ⊆ [a, b] using concept of variational measure. 2020 Mathematics Subject Classifications: 26A39, 26A42 Key Words and Phrases: McShane integral, integrable set, McShane δ-variation, McShane variational measure, variation zero, Cauchy extension. 1. Introduction Let f : [a, b]→ R be a function. It is well-known that f is McShane integrable on [a, b] if and only if f is Lebesgue integrable on [a, b] and the values of the integrals are equal, see [5, Theorem 10.11]. If f is McShane integrable on [a, b], then f is Henstock-Kurzweil integrable on [a, b] (with the same value of integrals) but the converse is not true, as seen in Example 1 below. If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on any sub-interval [c, d] of [a, b], see [5, Theorem 10.4], [6], [7] and [10]. It is shown in [8] that f : [a, b]→ R is Henstock-Kurzweil integrable on [a, b] if and only if for each c ∈ (a, b) the function f · χ[a,c] is Henstock-Kurzweil integrable on [a, c] and lim c→b− ∫ c a f exists. In this case, lim c→b− ∫ c a f = ∫ b a f. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3659 Email addresses: felipejr.sumalpong@g.msuiit.edu.ph (F. Sumalpong Jr.), julius.benitez@g.msuiit,edu.ph (J. Benitez) http://www.ejpam.com 303 c© 2020 EJPAM All rights reserved. F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 304 This is known as the Cauchy Extension. Example 1 also shows that Cauchy extension does not hold for McShane integral. Indeed, f is McShane integrable on [ε, 1] for every 0 < ε < 1 and lim ε→0+ ∫ 1 ε f = lim ε→0+ (F (1)− F (ε)) = sin 1, but f is not McShane integrable on [0, 1]. Example 1. Consider the function F : [0, 1]→ R defined as follows: F (x) = { x2 sin 1 x2 , if x 6= 0; 0 , if x = 0. Let f : [0, 1]→ R be the function defined by f(x) = { F ′(x) , if 0 < x ≤ 1; 0 , if x = 0, where F ′(x) = 2x sin 1 x2 − 2 x cos 1 x2 , for 0 < x ≤ 1. We observe that F is continuous on [0, 1]. Now, we show that F is not of bounded variation on [0, 1]; that is, V (F ; [0, 1]) =∞, where V (F ; [0, 1]) = sup D∈D (D) ∑ |F (v)− F (u)| and D is the class of all partition D = {[u, v]} of [0, 1]. Note that F (x) = ±x2 if and only if x2 = 2 2nπ±π , where n is a positive integer. For each n ∈ N, let xn = √ 2 2nπ+π . Then for each n ∈ N F (xn) = 2 · (−1)n 2nπ + π . Thus, ∞∑ n=0 |F (xn)− F (xn+1)| = ∞∑ n=0 ∣∣∣∣2 · (−1)n 2nπ + π − 2 · (−1)n+1 2nπ + 3π ∣∣∣∣ = 2 π ∞∑ n=0 4n+ 4 4n2 + 8n+ 3 ≥ 2 π ∞∑ n=0 1 n+ 1 = 2 π ∞∑ n=1 1 n =∞ that is, ∞∑ n=0 |F (xn)− F (xn+1)| =∞. Hence, V (F ; [0, 1]) ≥ ∞∑ n=0 |F (xn)− F (xn+1)|, F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 305 and so, V (F ; [0, 1]) =∞. Thus, F is not of bounded variation. Recall that in [4, p.19], if f : [a, b]→ R is McShane integrable on [a, b], then the primitive of f is of bounded variation. Hence, f is not Mc- Shane integrable on [0, 1]. Moreover, for each ε ∈ (0, 1), f is continuous on [ε, 1] which implies that f is McShane integrable on every closed interval [ε, 1]. However, by Theorem 16 of [11], f is Henstock integrable on [0, 1]. � If f is Henstock-Kurzweil integrable on every measurable subset of [a, b], then f is Lebesgue integrable on [a, b], see [5, Theorem 9.13]. In particular, if f is McShane inte- grable on every measurable subset of [a, b], then f is McShane integrable on [a, b]. It was pointed out in [3] that if f : [a, b]→ R is McShane integrable on [a, b], then f is McShane integrable on [a, b] on every integrable (equivalently, Lebesgue measurable) sub- set of [a, b]. If f is Henstock-Kurzweil integrable on [a, b], then [a, b] contains a subinterval on which f is Lebesgue (McShane) integrable, see [5, Corollary 9.19] and [8]. Thus, a natural problem is: If f : [a, b]→ R is McShane integrable on [a, b] and X ⊆ [a, b], find a condition satisfied by X which is necessary and sufficient for the McShane integrability of f on X. In this paper, we give a characterization for the McShane integrability of f : [a, b]→ R on X ⊆ [a, b] using concept of McShane variational measure. 2. Preliminary Concepts and Known Results A gauge on [a, b] is a positive function δ : [a, b] → R+. A Henstock δ-fine division of [a, b] is a finite collection D = {([xi−1, xi], ξi)}ni=1 of non-overlapping interval-point pairs such that for all i = 1, 2, . . . , n ξi ∈ [xi−1, xi] ⊆ (ξi − δ(ξi), ξi + δ(ξi)) and n⋃ i=1 [xi−1, xi] = [a, b]. We say that D = {([xi−1, xi], ξi)}ni=1 is a McShane δ-fine division of [a, b] if for all i = 1, 2, . . . , n [xi−1, xi] ⊆ (ξi − δ(ξi), ξi + δ(ξi)), ξi ∈ [a, b], and n⋃ i=1 [xi−1, xi] = [a, b]. This means that every Henstock δ-fine divisions of [a, b] are McShane δ-fine. For brevity, we use ([u, v], ξ) to represent a typical interval-point pair ([xi−1, xi], ξi) ∈ D. A finite collection P = {([u, v], ξ)} of interval-point pairs is a partial division of [a, b] if ∪[u, v] ⊆ [a, b]. Interested readers may refer to [5–7, 10] for more details on the basic concepts introduced. F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 306 Lemma 1 (Cousin’s Lemma). [6] If δ is a gauge on [a, b], then there exists a δ-fine division of [a, b]. Definition 1. [5–7, 10] A function f : [a, b] → R is said to be McShane integrable to a real number A on [a, b] if for any ε > 0, there exists a gauge δ : [a, b]→ R+ such that for any McShane δ-fine division D = {([u, v], ξ)} of [a, b], we have∣∣∣(D) ∑ f(ξ)(v − u)−A ∣∣∣ < ε. If f : [a, b]→ R is McShane integrable to A on [a, b], then we write A = ∫ b a f. For E ⊆ [a, b], we say that f : [a, b] → R is McShane integrable on E if f · χE is McShane integrable on [a, b], where χE is the characteristic function of E and write∫ E f = ∫ b a (f · χE). The next theorem tells us that the McShane integral is an absolute integral. Theorem 1. [7] If f : [a, b]→ R is McShane integrable on [a, b], then so is |f |. If f : [a, b]→ R is McShane integrable on [a, b], then the function F : [a, b]→ R defined by F (x) = ∫ x a f, for all x ∈ [a, b] is called the primitive of f . Almost all significant results in the Henstock Integration Theory rely on the following very important Theorem involving the primitive, called the Henstock’s Lemma. Theorem 2 (Henstock’s Lemma). [6] If f : [a, b] → R is McShane (resp., Henstock) integrable on [a, b] with primitive F , then for each ε > 0, there exist δ : [a, b] → R+ such that whenever D = {([u, v], ξ)} is a McShane (resp., Henstock) δ-fine division of [a, b], we have (D) ∑∣∣∣f(ξ)(v − u)− F (v) + F (u) ∣∣∣ < ε. The following concept was introduced by Yang [12]. Definition 2. A set E ⊆ R is integrable if χE∩[a,b] is McShane integrable on [a, b], for all [a, b] ⊆ R. Clearly, ∅ and R are integrable subsets of R. Moreover, it can be seen that the collection I of all integrable subsets of R is a σ-algebra. In [9], Quindala and Benitez showed that the set-function µ : I → [0,∞] defined by µ(E) = ∫ ∞ −∞ χE , for all E ∈ I F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 307 is a measure. In the same paper, for a bounded integrable set E ⊆ R µ(E) = m∗(E) where m∗ is the Lebesgue outer measure. See [1, p.85] for the proof of the following Lemma which utilizes the Heine-Borel Covering Theorem. Lemma 2. [1] If E ⊆ [a, b] and χE is McShane integrable on [a, b], then for all ε > 0 there exists an open set O ⊆ [a, b] such that E ⊆ O and∫ b a χOrE < ε. The following is a characterization of an integrable set E ⊆ [a, b]. Lemma 3. [12] Let E ⊆ [a, b]. The following statements are equivalent: (i) χE is McShane integrable on [a, b]. (ii) E is integrable. Definition 3. An integrable set E ⊆ [a, b] is said to have variation zero if∫ b a χE = 0. It is worth noting that a subset of a set of variation zero is again of variation zero. Definition 4. A property is said to hold almost everywhere (abbreviated a.e.) on A if the set of points in A where it fails to hold is an integrable set of variation zero. It was proved in [9] that the notions of McShane integrable set and Lebesgue mea- surable set are equivalent. This implies that integrable sets of variation zero are exactly those subsets of [a, b] with zero Lebesgue measure. Furthermore, the concept of “almost everywhere” in the sense of McShane, introduced in Definition 4, coincides with the corre- sponding concept from Lebesgue theory. This equivalences do not diminish the relevance of the notion of McShane integrable set. If functions possess certain properties almost everywhere, then some properties of the Henstock integral are preserved. In particular, if two functions are equal a.e. and one of the functions is McShane integrable, then the other function is also McShane integrable and their integral values coincide. This is precisely stated in Theorem 3. The proof of this result is standard and one may follow the proof of Theorem 9.5 in [5] or Theorem 10 in [11]. F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 308 Theorem 3. Let f : [a, b] → R be McShane integrable on [a, b]. If g = f a.e. on [a, b], then g is McShane integrable on [a, b], and∫ b a g(x) dx = ∫ b a f(x) dx. In view of Theorem 3, the condition “fn(x) → f(x) on [a, b]” in all convergence the- orems for the McShane integral (see [5]) can now be replaced by “fn(x) → f(x) a.e. on [a, b]”. Theorem 4. [4] If f : [a, b] → R is McShane integrable on [a, b], then there exists a sequence {ϕn}∞n=1 of step functions such that ϕn → f a.e. on [a, b] and lim n→∞ ∫ b a |ϕn − f | = 0. In the following, if f : [a, b]→ R is a function and c ∈ R, then we denote E(f < c) = {x ∈ [a, b] : f(x) < c}. Theorem 5. [2] Let c be any real number. If f : [a, b] → R is McShane integrable on [a, b], then the characteristic function χE(f c), or E(f ≤ c) then χA is McShane integrable on [a, b]. Hence, by Lemma 3 and Theorem 5, we have the following result: Corollary 1. Let c be any real number. If f : [a, b] → R is McShane integrable on [a, b], then the sets E(f < c), E(f ≥ c), E(f > c) and E(f ≤ c) are integrable sets. Below is a version of the Monotone Convergence Theorem (MCT) for McShane inte- grals. Theorem 6 (Monotone Convergence Theorem). [1, 5] Let {fn}∞n=1 be an increasing se- quence of McShane integrable functions on [a, b] such that lim n→∞ fn(x) = f(x) , for each x ∈ [a, b]. F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 309 If sup {∫ b a fn : n ∈ N } <∞, then f is McShane integrable on [a, b] and lim n→∞ ∫ b a fn = ∫ b a f. Recall that if f : [a, b]→ R is McShane integrable on [a, b], then f is McShane integrable on every sub-interval [c, d] of [a, b]. However, McShane integrability on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. One necessary condition is that a subset E ⊆ [a, b] must be an integrable set. Theorem 7. [3] If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on every integrable subset E of [a, b]. 3. Results In what follows, we denote the family of all sub-intervals of [a, b] by I([a, b]) and δ is a gauge on [a, b]. Definition 5. [8] Let F : I([a, b]) → R. For any subset X of [a, b], the McShane δ- variation of F on X is given by V (F,X, δ) = sup P∈P([a,b]) (P ) ∑ |F (u, v)|, where P([a, b]) is the collection of all McShane δ-fine partial division P = {([u, v], ξ)} of [a, b] with ξ ∈ X. The McShane variational measure of F on X is given by VMF (X) = inf { V (F,X, δ) : δ is a gauge on X } . If f : [a, b]→ R is McShane integrable on [a, b] with primitive F , then we write F (u, v) = F (v)− F (u), for any u ≤ v in [a, b]. Lemma 4. Let f : [a, b] → R be McShane integrable on [a, b] with primitive F and X ⊆ [a, b]. If f is McShane integrable on X, then VMF (X) <∞ and∫ b a |f · χX | = VMF (X). Proof. Let ε > 0. By Henstock Lemma, there exists a gauge δ1 on [a, b] such that (P ) ∑ |F (u, v)− f(ξ)(v − u)| < ε 3 whenever P = {([u, v], ξ)} is a McShane δ1-fine partial division of [a, b]. F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 310 Note that if f is McShane integrable on X ⊆ [a, b], then f · χX is McShane integrable on [a, b]. By Theorem 1, |f · χX | is McShane integrable on [a, b]. Again by Henstock Lemma, there exists a gauge δ2 on [a, b] such that (P ) ∑∣∣∣|(f · χX)(ξ)|(v − u)− ∫ v u |f · χX | ∣∣∣ < ε 3 whenever P = {([u, v], ξ)} is a McShane δ2-fine partial division of [a, b]. Let δ3 = min{δ1, δ2} and P = {([u, v], ξ)} be a McShane δ3-fine partial division of [a, b] such that ξ ∈ X. Then (P ) ∑ |F (u, v)| ≤ (P ) ∑ |F (u, v)− f(ξ)(v − u)|+ (P ) ∑ |f(ξ)(v − u)| < ε 3 + (P ) ∑ |f(ξ)(v − u)| ≤ ε 3 + (P ) ∑∣∣∣|f(ξ) · χX(ξ)|(v − u)− ∫ v u |f · χX | ∣∣∣ +(P ) ∑∫ v u |f · χX | < 2ε 3 + (P ) ∑∫ v u |f · χX | ≤ 2ε 3 + ∫ b a |f · χX |. This implies that VMF (X) ≤ V (F,X, δ3) ≤ 2ε 3 + ∫ b a |f · χX |. Since ε > 0 is arbitrary, we have VMF (X) ≤ ∫ b a |f · χX |. (1) By definition of VMF (X), there exists δ4(ξ) > 0 such that V (F,X, δ4) ≤ VMF (X) + ε 3 . Let δ0(ξ) = min{δ3(ξ), δ4(ξ)}, for all ξ ∈ [a, b]. Let P = {([u, v], ξ)} be any McShane δ0-fine partial division of [a, b] with ξ ∈ X. Suppose D = P ∪ P ′ is a McShane δ0-fine division of [a, b], where P ′ = {([u, v], ξ)} such that ξ /∈ X. Then (D) ∑ |f(ξ) · χX(ξ)|(v − u) = (P ) ∑ |f(ξ) · χX(ξ)|(v − u). Note that D is also a McShane δ0-fine partial division of [a, b]. Thus,∫ b a |f · χX | ≤ (D) ∑∣∣∣ ∫ v u |f · χX | − |f(ξ) · χX(ξ)|(v − u) ∣∣∣+ (D) ∑ |f(ξ) · χX(ξ)|(v − u) F. Sumalpong Jr., J. Benitez / Eur. J. Pure Appl. Math, 13 (2) (2020), 303-313 311 ≤ (D) ∑∣∣∣ ∫ v u |f · χX | − |f(ξ) · χX(ξ)|(v − u) ∣∣∣+ (P ) ∑ |f(ξ) · χX(ξ)|(v − u) < ε 3 + (P ) ∑ |f(ξ) · χX(ξ)|(v − u) = ε 3 + (P ) ∑ |f(ξ)|(v − u) ≤ ε 3 + (P ) ∑∣∣f(ξ)(v − u)− F (u, v) ∣∣+ (P ) ∑∣∣F (u, v) ∣∣ < 2ε 3 + (P ) ∑∣∣F (u, v) ∣∣ ≤ 2ε 3 + V (F,X, δ0). Hence, ∫ b a |f · χX | ≤ ε+ VMF (X). Again, since ε > 0 is arbitrary, we have∫ b a |f · χX | ≤ VMF (X). (2) Combining (1) and (2), we have ∫ b a |f · χX | = VMF (X). Theorem 8. Let f : [a, b] → R be McShane integrable on [a, b] with primitive F and X ⊆ [a, b]. Then f is McShane integrable on X if and only if VMF (X) <∞. In this case,∫ b a |f · χX | = VMF (X). Proof. Necessity follows from Lemma 4. Conversely, assume that VMF (X) <∞. For each n ∈ N, we let Xn = {x ∈ X : |f(x)| ≤ n}. Then each Xn is an integrable set, Xn ⊆ Xn+1 for all n and X = ∞⋃ n=1 Xn. By Theorem 7, f is McShane integrable on Xn for each n. By Lemma 4, VMF (Xn) <∞ and ∫ b a |f · χXn | = VMF (Xn). REFERENCES 312 Consider the sequence {fn}∞n=1, defined by fn = |f · χXn | for all n ∈ N. Since Xn ⊆ Xn+1 for each n, {fn}∞n=1 is an increasing sequence of McShane integrable functions on [a, b] and fn → |f · χX | pointwisely on [a, b]. Note that sup {∫ b a |f · χXn | : n ∈ N } = sup { VMF (Xn) : n ∈ N } ≤ VMF (X) <∞. By Theorem 6, |f · χX | is McShane integrable on [a, b] and∫ b a |f · χX | = lim n→∞ ∫ b a |f · χXn | = lim n→∞ VMF (Xn) = VMF (X). 4. Conclusion Let f : [a, b] → R to be McShane (Lebesgue) integrable on [a, b] with primitive F . It is shown in this paper that for f to be McShane integrable on X ⊆ [a, b], a necessary and sufficient condition is that the primitive must be of finite McShane variational measure on X, that is, VMF (X) <∞. The authors recommend that interested readers may investigate this study for non-absolute integrals (since McShane integrals are absolute). References [1] Benitez, J.V., Equi-Integrability ad Harnack Extensions for McShane and Henstock Integrals, Ph.D. Dissertation, MSU-Iligan Institute of Technology, 2012. [2] Benitez, J.V., Jamil, F.P. and Chew, T.S., MCShane Integrability and Egoroff’s Theorem, Matimyás Matematika, 32(2), 13-20, 2009. [3] Benitez, J.V., Integrable Set and Measurable Function, Mindanawan Journal of Math- ematics, 3(1), 2012. [4] Chew, T.S., The Riemann-type Integral that Includes Lebesgue-Stieltjes and Stochastic Integrals, Lecture Note, Chulalongkorn University, 2004. [5] Gordon, R., The integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, Amer. Math. Soc., 1994. [6] Lee, P.Y., Lanzhou Lectures on Henstock Integration, World Scientific, 1989. [7] Lee, P.Y. and Yýborný, R., The integral: an easy approach after Kurzweil and Hen- stock, Cambridge University Press, 2000. [8] Lee, T.Y., Henstock-Kurzweil Integration on Euclidean Spaces, World Scientific, vol- ume 12, 2011. REFERENCES 313 [9] Benitez, J.V. and Quindala III, K.M., On McShane Integrable Sets and Lebesgue Measure, International Journal of Mathematical Analysis, 9(3), 127-139, 2015. [10] Swartz, C., Introduction to Gauge Integrals, World Scientific, 2001. [11] Wells, J. , Generalizations of the Riemann Integral: An Investigation of the Henstock Integral, 2011. [12] Yang, C.H., Measure Theory and the Henstock integral, Academic Exercise, National University of Singapore, Singapore, 1996-97.