# Gaussian measures on non-separable spaces

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 $X$?

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A measure concentrated in a line... is that considered non-trivial? – Gerald Edgar Mar 9 '13 at 17:12
why is there an algebraic-geometry tag? (it's not a complaint, it's curiosity) – Jacob Bell Mar 9 '13 at 21:48
@Gerald Edgar: good point. We can always construct measures which has support on a finite-dimensional subspace of $X$. Let's add the condition that the support of the measure is the whole space. – Tom LaGatta Mar 10 '13 at 20:31
@Jacob Bell: I suspect that algebraic geometers are the mathematicians who think the most about general categories of spaces like topological affine spaces. I put the tag to get their input. – Tom LaGatta Mar 10 '13 at 20:32
What does "gaussian measure" mean if there are no continuous linear functionals on the space? – Gerald Edgar Mar 10 '13 at 20:51

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.

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Thanks, @Anatoly Kochubei! Is is the case that in the non-separable setting, Gaussian measures are always concentrated on separable subspaces? It would seem reasonable that the answer is yes, which provides a negative answer to my original question. I was mulling over your second fact last night (the canonical Gaussian cylinder-set measure cannot be extended to a genuine measure), and am glad to see the reference. I'll take a look at Satô's paper and let you know if I've got any questions. – Tom LaGatta Mar 11 '13 at 11:10
Indeed, this does answer the question. The Corollary to Theorem 2 states that there is no admissible norm on a non-separable Hilbert space. Since the support of the measure in affine space $X$ is the closure of the Cameron-Martin space corresponding to the covariance operator, the support of the measure must be separable. Consequently, there can be no Gaussian measure with full support in a non-separable affine space. Cheers. – Tom LaGatta Mar 11 '13 at 11:25