Let $\mu$ be a Radon probability measure on a type 2 Banach space $E$. The following Bochner's theorem is proved. For every continuous positive definite function $\phi(\phi(0) = 1)$ on $E$, there exists a Radon probability measure $\sigma_\phi$ on the measurable dual $H_0(\mu)$ of $(E, \mu)$ with the characteristic functional $\phi$ (in some restricted sense).
"Bochner's Theorem in Measurable Dual of Type 2 Banach Space." Ann. Probab. 13 (3) 1022 - 1023, August, 1985. https://doi.org/10.1214/aop/1176992925