Fourier series expansion of stochastic measures

We consider processes of the form $\mu(t)=\mu((0,t])$, where $\mu$ is a $\sigma$-additive in probability stochastic set function. Convergence of a random Fourier series to $\mu(t)$ is proved, and the approximation of integrals with respect to $\mu$ using Fejèr sums is obtained. For this approximation, we prove the convergence of solutions of the heat equation driven by $\mu$.