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August, 1981 Gaussian Measurable Dual and Bochner's Theorem
Hiroshi Sato
Ann. Probab. 9(4): 656-662 (August, 1981). DOI: 10.1214/aop/1176994371


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 $\mu$.


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Hiroshi Sato. "Gaussian Measurable Dual and Bochner's Theorem." Ann. Probab. 9 (4) 656 - 662, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0464.60017
MathSciNet: MR624692
Digital Object Identifier: 10.1214/aop/1176994371

Primary: 60E10
Secondary: 28C20 , 60B11

Keywords: Bochner's theorem , characteristic functional , Gaussian-Radon measure , measurable dual

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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