Bochner's theorem formulated by Xia Dao-Xing is established for an abstract Wiener space. Let $(\iota, H, E)$ be an abstract Wiener space. Then for every continuous cylinder set measure $\nu$ on $E'$, the image $\iota'(\nu)$ is a Radon measure on $H'$.
"Bochner's Theorem on Measurable Linear Functionals of a Gaussian Measure." Ann. Probab. 9 (4) 663 - 664, August, 1981. https://doi.org/10.1214/aop/1176994372