MEASURABLE FUNCTIONS ON HILBERT SPACE

extends in a well known way to a measure v on the Borel sets of L, which is independent of the choice of basis yt,---,y„ used in constructing it and depends only on the mapping F. Every probability Borel measure on L arises in this way from such a linear mapping F for if co is a probability Borel measure then the elements of L* are measurable functions on the probability space (L, co) and the identity map on L* fulfills the role of F, inducing as above a measure v which may easily be seen to coincide with a>. Clearly two such linear mappings F and F' induce the same measure v on L if and only if for each finite set yu ••-,)'* of elements of L* (in particular for a basis of L*) the random variables £0>i), •••,£(.)>*) on the one hand and F'CFi). •••,-F'O'*) on the other hand have the same joint distribution in fc-space. This correspondence between linear mappings F and probability measures v on L breaks down in case L is not finite dimensional. If L is, say, an infinite dimensional locally convex topological linear space then it is still true that a reasonable probability measure on L gives rise to a linear mapping F from the dual space L* of continuous linear functionals on L, to random variables, namely the identity mapping on L*. However a given linear map from L* to random variables will not in general induce a measure on L due to the absence of local compactness of L. By a random variable on a probability space (S, Sf, p.) we shall mean an element of the linear space of measurable functions modulo null functions on S. Definition. A weak distribution on a topological linear space L is an equivalence class of linear maps F from L* to random variables on a probability space (S, £P, p) (depending on F) where two such maps F and F' are equivalent