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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Zero-one laws for Gaussian measures on Banach space
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by Charles R. Baker PDF
Trans. Amer. Math. Soc. 186 (1973), 291-308 Request permission

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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 186 (1973), 291-308
  • MSC: Primary 60G15; Secondary 28A40
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0336796-5
  • MathSciNet review: 0336796