Throughout this paper Ε denotes a locally convex Hausdorff vector space over the field of real numbers (lcs). A Radon probability measure μ on Ε is said to be a (centred) Gaussian Radon measure on Ε if the image measure ξ (μ) is a (centred) Gaussian measure on R for every ξ belonging to the topological dual E'of E. The class of all (centred) Gaussian Radon measures on Ε is denoted by & (E)(@ 0 (E)). In Section 2, it will be proved that every μ e@(E) has a reproducing kernel Hilbert space (RKHS) 34?(μ) contained in E. One of the main results of this paper shows that 3ί?(μ) is separable (Theorem 7.1). This conclusion has many corollaries. For example, it follows that Σρ (μ)(l? p<+ oo) and supp (/�) are separable (Corollaries 8.1 and 8.2). Theorem 2.1 shows that every μ e@(E) has a barycentre be E. Setting μ0 (�)= μ (�+ b), it follows that 3?(μ) and the closure Ε2'(μ) of E'in Ιι2 (μο) are isomorphic (Theorem 2.1). This makes it possible to give a very simple representation of^-measurable additive functions. A real valued function/on Ε is said to be a/�-measurable additive (subaddi tive) function on E, if/(+�) are/�-measurable, and there exists an addi tive/�-measurable subgroup G of Ε with/�-measure one so that f (x+ y)=(�) f (x)+ f (y), x, yeG.

Since Ε2'(μ) is separable, there exists an at most denumerable ortho normal basis {en} for this Hilbert space so that every en belongs to E'. If/is a/immeasurable additive function on E, it will be proved that there exists an (an) e l2 (N) such that