Equimeasurable sets and cylindrical measures

We obtain a characterization of equimeasurable sets in the space $S(\Omega ,\Sigma,\mathbf{P})$ in terms of the coincidence of convergence in probability and almost sure convergence. The notion of an equimeasurable set is used to obtain criteria for extending a cylindrical measure to a Radon measure and also to establish a criterion of the existence of continuous trajectories of a linear random function on an absolutely convex weak compact set.