The following generalizations of certain theorems due to G. Kallianpur and to Jamison and Orey are proved for an arbitrary Gaussian measure $P$ on a space of real functions: if the reproducing kernel Hilbert space $H$ is infinite dimensional then $P(H) = 0$; if a subgroup $G$ of the space of real functions (under addition) is measurable with respect to the $P$-completion of the Borel product sigma-algebra, then $P(G) = 0$ or $P(G) = 1$ and in the latter case $H \subset G$.