Carriers of continuous measures in a Hilbertian norm

Abstract

From the standpoint of the theory of measures on the dual space of a nuclear space, we discuss the carrier of Wiener measure, regarding it as a measure on (${\mathcal{D}}^{\prime }$) (= Schwartz's space of distributions). This may be contrasted with the usual treatment which regards it as a measure on the space of paths.

It is shown that for $\alpha >\frac{1}{2}$, integral operator $I_{\alpha }$ is nuclear on $L^2((0,1))(\equiv H_0)$. Using this fact, we see that Wiener measure lies on the space $I_{\beta }(H_0)(\beta <\frac{1}{2})$ which consists of Holder continuous functions of the $\beta$-th order in the sense of $L^2$. This result is true for any measure whose characteristic functional is continuous on $L^2((0,1))$.