In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is $F_{\sigma \delta }$ regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.
It is shown that if g is of bounded variation in the sense of Hardy-Krause on ∏m i=1 [ai , bi ], then gχ ∏m i=1 (ai ,bi ) is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.
It is shown that a Banach-valued Henstock-Kurzweil integrable function on an m-dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function f : [0, 1]2 −→ R and a continuous function F : [0, 1]2 −→ R such that
(P)∫ x0{ (P) ∫ y 0 f(u, v) dv } du = (P) ∫ y 0 { (P) ∫ x 0 f(u, v) du } dv = F(x, y) for all (x, y) ∈ [0, 1]2.
Applying a simple integration by parts formula for the Henstock-Kurzweil integral, we obtain a simple proof of the Riesz representation theorem for the space of Henstock-Kurzweil integrable functions. Consequently, we give sufficient conditions for the existence and equality of two iterated Henstock-Kurzweil integrals.
Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.
We establish two new norm convergence theorems for Henstock-Kurzweil integrals. In particular, we provide a unified approach for extending several results of R. P. Boas and P. Heywood from one-dimensional to multidimensional trigonometric series.