Let $X$ be a topological affine space which is neither separable nor metrizable. There are plenty of trivial Gaussian measures: each Dirac point-mass $\delta_x$ are the Gaussian measure with zero covariance and mean $x \in X$

When does a topological affine space $X$ admit a non-trivial Gaussian measure? Namely, one whose support is the whole space $X$? One can always construct a Gaussian measure supported on a finite-dimensional subspace of $X$, such as a line.

Is local convexity of $X$ a sufficient criterion for there to exist a Gaussian measure supported on all of $X$?