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 $X$?



Gaussian measures on non-separable spaces
There is a paper by H. Sato, <http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795>


proving that 1) a Gaussian measure on a reflexive Banach space is always concentrated on a separable subspace, and 2) the canonical Gaussian cylinder measure of a nonseparable Hubert space cannot be extended to a $\sigma$-additive measure in any Banach space.


