Bochner's Theorem in Measurable Dual of Type 2 Banach Space

Abstract

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).