## The Annals of Probability

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

### Gaussian Measurable Dual and Bochner's Theorem

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60E10: Characteristic functions; other transforms

Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60B11: Probability theory on linear topological spaces [See also 28C20]

**Keywords**

Measurable dual Bochner's theorem Gaussian-Radon measure characteristic functional

#### Citation

Sato, Hiroshi. Gaussian Measurable Dual and Bochner's Theorem. Ann. Probab. 9 (1981), no. 4, 656--662. doi:10.1214/aop/1176994371. https://projecteuclid.org/euclid.aop/1176994371