We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by ${\sum }_{n=1}^{\infty }{x}_n{\chi }_{E_n}$, where $x_n$ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series ${\sum }_{n=1}^{\infty }{x}_n\left|E_n\right|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$....

We study properties of variational measures associated with certain conditionally convergent integrals in ${ℝ}^m$. In particular we give a full descriptive characterization of these integrals.

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a $𝒫$-adic path system that defines a differentiation basis which does not possess Ward property.

In this paper, we prove the existence of continuous solutions of a Volterra integral inclusion involving the Henstock-Kurzweil-Pettis integral. Since this kind of integral is more general than the Bochner, Pettis and Henstock integrals, our result extends many of the results previously obtained in the single-valued setting or in the set-valued case.