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 (D') (= 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 α > 1/2, integral operator Iα is nuclear on L2((0,1)) (≡H0). Using this fact, we see that Wiener measure lies on the space Iβ(H0) (β<1/2) which consists of Holder continuous functions of the β-th order in the sense of L2. This result is true for any measure whose characteristic functional is continuous on L2((0,1)).
Cite this article
Yasuo Umemura, Carriers of continuous measures in a Hilbertian norm. Publ. Res. Inst. Math. Sci. 1 (1965), no. 1, pp. 49–54