The natural modification of a random process and its applications to random functional series and Gaussian measures

A Gaussian random process is known to be expanded into a series of functions with random independent coefficients. Previded the process is continuous in the mean but not sample continuous, the corresponding series does not converge uniformly. In what cases does it converge pointwise? This question is reduced to well studied problem of the sample boundedness. It is showed that the pointwise convergence of expancion mentioned above is equivalent to the sample continuity of the process in some separable metric. Certain other properties of Gaussian processes and measures are considered, generalisations to a non-Gaussian case are given.