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.
A stochastic heat equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure $d\mu (t)$ is investigated in this paper. For the integrator μ, we assume the σ-additivity in probability only. The existence, uniqueness, and Hölder regularity of the solution are proved.