In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.

It is known that there is no natural Banach norm on the space $\mathrm{\mathscr{H}𝒦}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathrm{\mathscr{H}𝒦}$ space is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. On each $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space, an $F$-norm ${\parallel ·\parallel }^X$ is defined. A ${\parallel ·\parallel }^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a ${\parallel ·\parallel }^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is well-known that every control-convergent...

The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in ${ℝ}^n$ given by Kurzweil and...

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a $\sigma$-finite outer regular quasi Radon measure space $(S,\Sigma ,𝒯,\mu )$ into a Banach space $X$ and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function $f$ from $S$ into $X$ is weakly McShane integrable on each measurable subset of $S$ if and only if...

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.