On seminorms and probabilities, and abstract Wiener spaces

Abstract

In real Hilbert space H there is a finitely additive measure n on the ring of sets defined by finitely many linear conditions, which is analogous to the normal distribution in the finite-dimensional case. This has been examined from various points of view in Gelfand and Vilenkin [11], Gross [12], Segal [19], as well as by many earlier authors. Now, if the Hilbert space is “completed” in any of number of ways, this “cylinder set measure” extends to an actual Borel measure on the completed space. For example, if we define | x | _T = P Tx P, where T is a Hilbert-Schmidt operator with trivial nullspace, then the completion of H with respect to the norm | • | _T is such a completion. A more subtle example is the following. Let H be realized explicity as L ₂(0, 1). Define a norm on H by $|f|={\rm sup}_{0\leqq t \leqq 1}|\left|\int^{{t}}_{0}f\left(s\right)ds\right|.$ Then the completion of H with respect to this is isomorphic to the continuous functions on [0, 1] with suo norm and vanishing at zero, via the map which sends f to the function g whose value at t is $\int^{{t}}_{0}f\left(s\right)ds, 0\leqq t \leqq 1.$ The finitely additive measure n is then realized as Wiener measure. Other natural norms give rise to realizations of n as Wiener measure on the Hölder–continuous function, exponent α < α < 1/2. This sort of phenomenon has been investigated abstractly by L. Gross in [14]. He shows there (Theorem 4) that if H is completed with respect to any of his measurable seminorms, as defined in Gross [13], then n gives rise to a country additive Borel measure on the Banach space obtained frome H by means of the seminorm.