Moore-Smith Limits and the Henstock Integral

Abstract

An integral is defined using the Moore-Smith limit and this new integral is compared to the Henstock integral.It is well-known though not easily found in the literature that the Riemann integral can be defined by Moore-Smith limit using divisions. Then many properties of the Riemann integral will have straightforward proofs. In this paper, we shall investigate whether the Henstock integral can also be defined by means of Moore-Smith limit involving δ-fine divisions. We assume that the reader is familiar with the definition of the Henstock integral [3].A division D of [a, b] is a finite set of interval-point pairs ([u, v], ξ) such that the intervals [u, v] from D are non-overlapping and their union is [a, b] Then D 2 is said to be finer than D 1 in the Riemann sense, or in symbols, D 2 D 1 if for each ([s, t], η) ∈ D 2 we have [s, t] ⊂ [u, v] for some ([u, v], ξ) ∈ D 1 and when [s, t] = [u, v] we have η = ξ. Then (D, ) is a directed set of divisions D of [a, b] . More precisely, the following conditions are satisfied:

Integration with One-Dimensional Space of Gauges

“…The H 1 -integral has been introduced by Garces, Lee, and Zhao [3], in an attempt to define an integral with nearly the Kurzweil-Henstock integral power, but in terms of Moore-Smith limits. Later advances in the theory of H 1 -integration have shown that this challenge was not successful in some sense.…”

The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$

“…23-27, pp. 113-138] and [3], [8], [9]. In other words, the integral is defined by way of refinements of partitions and the integral is the Moore-Smith limit of the Riemann-Stieltjes sums using the directed set of partitions.…”