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 $\prod _{i=1}^{m}[{a}_{i},{b}_{i}]$, then $g{\chi}_{{}_{\prod _{i=1}^{m}({a}_{i},{b}_{i})}}$ is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula.

Several new integrability theorems are proved for multiple cosine or sine series.

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.

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.

We use an elementary method to prove that each $BV$ function is a multiplier for the $C$-integral.

Some full characterizations of the strong McShane integral are obtained.

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 $f\phantom{\rule{0.222222em}{0ex}}{[0,1]}^{2}\u27f6\mathbb{R}$ and a continuous function $F\phantom{\rule{0.222222em}{0ex}}{[0,1]}^{2}\u27f6\mathbb{R}$ such that $$\left(\P \right){\int}_{0}^{x}\left\{\left(\P \right){\int}_{0}^{y}f(u,v)\mathrm{d}v\right\}\mathrm{d}u=\left(\P \right){\int}_{0}^{y}\left\{\left(\P \right){\int}_{0}^{x}f(u,v)\mathrm{d}u\right\}\mathrm{d}v=F(x,y)$$
for all $(x,y)\in {[0,1]}^{2}$.

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $({\mathcal{S}}_{\rho}\left(E\right),\parallel \xb7\parallel )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel \xb7\parallel $. We show that each element in the dual space of $({\mathcal{S}}_{\rho}\left(E\right),\parallel \xb7\parallel )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere...

Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.

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