We present a descriptive definition of a multidimensional generalized Riemann integral based on a concept of generalized absolute continuity for additive functions of sets of bounded variation.

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.

A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from . The theory is recovered together with a few new results.

A descriptive characterization of a Riemann type integral, defined by BV partition of unity, is given and the result is used to prove a version of the controlled convergence theorem.

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho$-integral, introduced by Jarník and Kurzweil. Let $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ be the space of all strongly $\rho$-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel ·\parallel$. We show that each element in the dual space of $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ 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...

We present a Cauchy test for the almost derivability of additive functions of bounded BV sets. The test yields a full descriptive definition of a coordinate free Riemann type integral.

This paper presents a Komlós theorem that extends to the case of the set-valued Henstock-Kurzweil-Pettis integral a result obtained by Balder and Hess (in the integrably bounded case) and also a result of Hess and Ziat (in the Pettis integrability setting). As applications, a solution to a best approximation problem is given, weak compactness results are deduced and, finally, an existence theorem for an integral inclusion involving the Henstock-Kurzweil-Pettis set-valued integral is obtained.

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.

The paper is concerned with integrability of the Fourier sine transform function when $f\in {\mathrm{BV}}_0\left(ℝ\right)$, where ${\mathrm{BV}}_0\left(ℝ\right)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f/x\in L^1\left(ℝ\right)$. We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

It is shown that there exist a continuous function $f$ and a regulated function $g$ defined on the interval $[0,1]$ such that $g$ vanishes everywhere except for a countable set, and the $K^{*}$-integral of $f$ with respect to $g$ does not exist. The problem was motivated by extensions of evolution variational inequalities to the space of regulated functions.

Integration by parts results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration based on Riemann type integral sums, which leads to the Perron integral.