International Journal of Mathematical Analysis Vol. 8, 2014, no. 51, 2539-2545 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/10.12988/ijma.2014.410302 ACG M and ACG H Functions Julius V. Benitez Department of Mathematics and Statistics College of Science and Mathematics MSU-Iligan Institute of Technology 9200 Iligan City, Lanao del Norte, Philippines Copyright c 2014 Julius V. Benitez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract In this paper, we introduce the ACG M and ACG H show that they are equivalent to the ACG property. properties and Mathematics Subject Classification: 26A46 Keywords: ACG, ACG M, ACG H, McShane δ-fine, Henstoc δ-fine 1 Introduction The ACG property is used in introducing the Denjoy integral. In his attempt to overcome the difficulty of the Denjoy integral to attain a natural generalization to higher dimensions, Lee [1] introduced the ACG property. In [3], it is shown that on the real line, ACG and ACG are equivalent. Given a positive function δ(ξ) > 0 on [a, b], a division or partial division D = {([u, v]; ξ)} of [a, b] is McShane δ-fine if [u, v] (ξ δ(ξ), ξ + δ(ξ)). If, in addition, ξ [u, v], then D = {([u, v]; ξ)} is Henstoc δ-fine. The Mc- Shane δ-fine and Henstoc δ-fine divisions are used in defining the McShane and Henstoc integrals, respectively. In both definitions, the function δ(ξ) > 0 is called a gauge. It is well-nown that the Henstoc integral is equivalent to the Denjoy integral, while the McShane integral is equivalent to the Lebesgue
2540 Julius V. Benitez integral. In this paper, we shall investigate the ACG property when defined in terms of the McShane and Henstoc δ-fine divisions or partial divisions of [a, b]. 2 ACG Functions The classical definition of the ACG property is given in [1]. However, in this paper, unless specified, we adopt the alternative definition used in [2] which has simplified the proof of the equivalence of the Henstoc and Denjoy integrals as exhibited in [3]. For continuous functions, these two versions are equivalent. The following is the classical definition of the ACG. Definition 2.1 [1] Let X [a, b]. A function F : [a, b] R is said to be AC (X) if for every ɛ > 0, there exists η > 0 such that for every finite or infinite sequence {[a i, b i ]} of non-overlapping intervals with a i, b i X b i a i < η implies ω(f ; [a i, b i ]) < ɛ, i where ω(f ; [a i, b i ]) = sup{ F (y) F (x) : x, y [a i, b i ]} is called the oscillation of F over [a i, b i ]. The function F is said to be ACG on [a, b] if [a, b] is the union of X, = 1, 2,..., such that F is AC (X ) for each. Next, we give the alternative definition which is due to Lee and Yyborny. Definition 2.2 [2] Let X [a, b]. A function F : [a, b] R is said to be AC (X) if for every ɛ > 0, there exists η > 0 such that for any partial division P = {[u, v]} with u or v X (P ) v u < η implies (P ) F (v) F (u) < ɛ. The function F is said to be ACG on [a, b] if [a, b] is the union of X, = 1, 2,..., such that F is AC (X ) for each. If F : [a, b] R is continuous, then Definition 2.1 and Definition 2.2 are equivalent [1]. Definition 2.3 [3] Let X [a, b]. A function F : [a, b] R is said to be AC (X) if for every ɛ > 0 there exist η > 0 and a gauge function δ(ξ) > 0 on [a, b] such that for any two Henstoc δ-fine partial divisions D = {([u, v], ξ)} and D = {([u, v ], ξ )} with D D (any subinterval in D lies in some interval in D) with ξ, ξ X, we have (D D ) v u < η implies (D D ) F (v) F (u) < ɛ i
ACG M and ACG H functions 2541 where (D D ) means (D) (D ). More precisely, D D consists of all ([u, v] [u, v ], ξ) whenever [u, v ] [u, v]. We say that F : [a, b] R is ACG if [a, b] is the union of a sequence of sets {X } =1 such that on each X the function F is AC (X ). D in Definition 2.3 may be void. Note, further, that every ([u, v] [u, v ], ξ) D D is McShane δ-fine with tag point ξ D. Lee [3] showed that among continuous functions on [a, b], ACG and ACG are equivalent. Theorem 2.4 [3] Let F be continuous on [a, b]. Then F is ACG if and only if F is ACG. Now, we define ACG in the sense of McShane δ-fine divisions. Definition 2.5 Let X [a, b]. A function F : [a, b] R is said to be ACM (X) if for every ɛ > 0 there exist η > 0 and a gauge δ(ξ) > 0 on [a, b] such that for any McShane δ-fine partial division P = {([u, v], x)} with x X, we have (P ) v u < η implies (P ) F (v) F (u) < ɛ. We say that F : [a, b] R is ACG M if [a, b] is the union of a sequence of sets {X } =1 such that on each X the function F is ACM (X ). are equivalent prop- The following theorem shows that ACG and ACG M erties. Theorem 2.6 Let F be a real-valued function on [a, b]. Then F is ACG M if and only if F is ACG. Proof : Assume that [a, b] = n=1 X n and F is ACM (X n) for each n. Let ɛ > 0. Then there exist η > 0 and a gauge δ(x) > 0 on [a, b] such that whenever P = {([u, v], x)} is a McShane δ-fine partial division of [a, b] with x X n, (P ) v u < η implies (P ) F (v) F (u) < ɛ. Let P and P be Henstoc δ-fine partial divisions of [a, b] whose tag points belong to X n with P P and (P P ) v u < η. Note that P P is a set of McShane δ-fine interval-point pairs ([u, v], x) with x a tag in P (and so, x X n ). Hence, (P P ) F (v) F (u) < η.
2542 Julius V. Benitez Thus, F is AC (X n ). Conversely, suppose that [a, b] = n=1 X n and F is AC (X n ) for each n. Let ɛ > 0. Then there exist η > 0 and a gauge δ(x) > 0 on [a, b] such that whenever P and P are Henstoc δ-fine partial divisions of [a, b] whose tag points belong to X n with P P, we have (P P ) v u < η implies (P P ) F (v) F (u) < ɛ. Let P = {([u, v], x)} be a McShane δ-fine partial division with x X n and Define the following: P 1 = {([u, v], x) P : u < v < x}, (P ) v u < η. P 2 = {([u, v], x) P : x < u < v}, P 1 = {([u, x], x) : ([u, v], x) P 1 }, P 2 = {([x, v], x) : ([u, v], x) P 2 }, P 1 = {([v, x], x) : ([u, v], x) P 1 }, P 2 = {([x, u], x) : ([u, v], x) P 2 }, and P 3 = {([u, v], x) P : u x v} Then P 1, P 1, P 2 and P 2 are Henstoc δ-fine divisions of [a, b] and v u = (P 1 ) v u (P ) v u < η and Hence, and Thus, [u,v ] P 1 P 1 [u,v ] P 2 P 2 v u = (P 2 ) v u (P ) v u < η. (P 1 ) F (v) F (u) = (P 2 ) F (v) F (u) = [u,v ] P 1 P 1 [u,v ] P 2 P 2 F (v ) F (u ) < ɛ F (v ) F (u ) < ɛ. (P ) F (v) F (u) = (P 1 ) F (v) F (u) + (P 2 ) F (v) F (u) < 3ɛ. +(P 3 ) F (v) F (u) This shows that F is AC M (X n). Next, we give the Henstoc counterpart of Definition 2.5.
ACG M and ACG H functions 2543 Definition 2.7 Let X [a, b]. A function F : [a, b] R is said to be ACH (X) if for every ɛ > 0 there exist η > 0 and δ(ξ) > 0 on [a, b] such that for any Henstoc δ-fine partial division P = {([u, v], x)} with x X, we have (P ) v u < η implies (P ) F (v) F (u) < ɛ. We say that F : [a, b] R is ACG H if [a, b] is the union of a sequence of sets {X } =1 such that on each X the function F is ACH (X ). It is quite apparent that if X X and F is AC H (X), then F is AC H (X ). Further, since Henstoc δ-fine partial divisions are also McShane δ-fine partial division, we have the following remar. Remar 2.8 Let F be a real-valued function on [a, b]. If F is ACG M, then F is ACG H. Lemma 2.9 Let F be a real-valued function on [a, b]. If F is ACG, then F is ACG H. Proof : Suppose that [a, b] = n=1 X n and F is AC (X n ) for each n. Let ɛ > 0. Then there exists η > 0 such that whenever P = {[u, v]} is a partial division of [a, b] with u or v X n and (P ) v u < η, we have (P ) F (v) F (u) < ɛ 2. Tae any δ(ξ) > 0 on X n, e.g. δ(ξ) = 1 for all ξ X n, and let P = {([u, v], x)} be any Henstoc δ-fine division of [a, b] with x X n and (P ) v u < η. Since x X n and (P ) x u (P ) v u < η, we have Similarly, (P ) F (x) F (u) < ɛ 2. (P ) F (v) F (x) < ɛ 2. Hence, (P ) F (v) F (u) (P ) F (v) F (x) + (P ) F (x) F (u) < ɛ 2 + ɛ 2 = ɛ. Thus, F is AC H (X n).
2544 Julius V. Benitez Theorem 2.10 Let F : [a, b] R be a continuous function. Then F is ACG if and only if F is ACG H. Proof : The necessity part has already been established by Lemma 2.9. Assume [a, b] = n=1 X n and F is ACH (X n) for each n. Let ɛ > 0. For each n, there exist η n > 0 and δ n (x) > 0 on [a, b] such that whenever P = {([u, v], x)} is a Henstoc δ n -fine partial division of [a, b] with x X n and (P ) v u < η n, we have (P ) F (v) F (u) < ɛ. We may assume that 0 < δ n (x) 1 for all n N. Let { 1 X n,i,j = x X n : < δ i+1 n(x) 1 j 1, x [a + }., a + j ] i i+1 i+1 For every n, X n,i,j X n for each i, j N. We claim that F is AC (X n,i,j ), and proceed using Definition 2.1. Now, fix X n,i,j and let P = {[a, b ]} be any partial division of [a, b] such that a, b X n,i,j and (P ) b a < η n. Note that for the partial division {([a, b ], a )}, we have b a (a + j j 1 ) (a + ) = 1 < δ i+1 i+1 i+1 n(a ), that is, {([a, b ], a )} is Henstoc δ n -fine partial division. Hence, F (b ) F (a ) < ɛ. Furthermore, if a u v b, then {([a, u ], a )} and {([v, b ], b )} are Henstoc δ n -fine partial divisions of [a, b] and v b b a < η n and a u b a < η n, implying that F (v ) F (b ) < ɛ and F (a ) F (u ) < ɛ. Thus, F (v ) F (u ) F (v ) F (b ) + F (b ) F (a ) + F (a ) F (u ) < ɛ + ɛ + ɛ = 3ɛ.
ACG M and ACG H functions 2545 Hence, ω(f ; [a, b ]) = ( sup { F (v ) F (u ) : u, v [a, b ] }) { } = sup F (v ) F (u ) : u, v [a, b ] < 3ɛ. This shows that F is AC (X n,i,j ). Therefore, F is ACG. In view of Theorem 2.4, Theorem 2.6 and Theorem 2.10, the following corollary follows: Corollary 2.11 Let F be continuous on [a, b]. Then the following statements are equivalent: i. F is ACG. ii. F is ACG M. iii. F is ACG H. References [1] Lee, Peng Yee, Lanzhou Lectures on Henstoc Integration, World Scientific, 1989. http://dx.doi.org/10.1142/0845 [2] Lee, Peng Yee and Yyborny, R., The integral: an easy approach after Kurzweil and Henstoc, Cambridge University Press, 2000. [3] Lee, Peng Yee, On ACG Functions, Real Analysis Exchange, 15(2) (1989/90), 754-759. Received: October 15, 2014; Published: November 20, 2014