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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Zero-one laws for Gaussian measures on Banach space

Author: Charles R. Baker
Journal: Trans. Amer. Math. Soc. 186 (1973), 291-308
MSC: Primary 60G15; Secondary 28A40
MathSciNet review: 0336796
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Abstract: Let $ \mathcal{B}$ be a real separable Banach space, $ \mu $ a Gaussian measure on the Borel $ \sigma $-field of $ \mathcal{B}$, and $ {B_\mu }[\mathcal{B}]$ the completion of the Borel $ \sigma $-field under $ \mu $. If $ G \in {B_\mu }[\mathcal{B}]$ is a subgroup, we show that $ \mu (G) = 0$ or 1, a result essentially due to Kallianpur and Jain. Necessary and sufficient conditions are given for $ \mu (G) = 1$ for the case where G is the range of a bounded linear operator. These results are then applied to obtain a number of 0-1 statements for the sample function properties of a Gaussian stochastic process. The zero-one law is then extended to a class of non-Gaussian measures, and applications are given to some non-Gaussian stochastic processes.

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Keywords: Gaussian measures, Gaussian stochastic processes, zero-one laws
Article copyright: © Copyright 1973 American Mathematical Society