Random functions $\mu (x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu (x)$, $x\in {[0,1]^{d}}$, is proved. Fourier series expansion of $\mu (x)$, $x\in [0,2\pi ]$, is obtained. These results are proved under weaker conditions than similar results in previous papers.
The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.