## The Annals of Probability

### Gaussian Measurable Dual and Bochner's Theorem

Hiroshi Sato

#### Abstract

Let $E$ be a locally convex Hausdorff linear topological space, $E'$ be the topological dual of $E$ and $\gamma$ be a nondegenerate, centered Gaussian-Radon measure on $E$. Then every nonnegative definite continuous functional on $E$ is the characteristic functional of a Borel probability measure on $E^\gamma$, the closure of $E'$ in $L_0(\gamma)$. In other words, identifying $E^\gamma$ with the reproducing kernel Hilbert space $\mathscr{H}_\gamma$ of $\gamma$, we may say that for every continuous nonnegative definite function $f$ on $E$ there exists a Borel probability $\mu$ on $\mathscr{H}_\gamma$ such that $f$ is the characteristic functional of $\mu$.

#### Article information

Source
Ann. Probab., Volume 9, Number 4 (1981), 656-662.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994371

Digital Object Identifier
doi:10.1214/aop/1176994371

Mathematical Reviews number (MathSciNet)
MR624692

Zentralblatt MATH identifier
0464.60017

JSTOR