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June, 1972 A Note on the Zero-one Law
Julius R. Blum, Pramod K. Pathak
Ann. Math. Statist. 43(3): 1008-1009 (June, 1972). DOI: 10.1214/aoms/1177692564


Let $\mathscr{M} = \{\mu_n: n \geqq 1\}$ be a sequence of probability measures defined on the measurable space $(\mathscr{R}_n, \mathscr{B}_n)$ and suppose that the measures $\{\mu_n: n \geqq 1\}$ satisfy the following condition $(\mathbf{B}): \mathbf{\forall}_\varepsilon > 0, k \geqq 1$ and $m \geqq 1$, there exists an $n \geqq m$ such that $\|\mu_k - \mu_n\| < \varepsilon$. We show that if $A \in \times^\infty_1 \mathscr{B}_n$ and if $A$ is permutation invariant then $\mu(A) = 0$ or 1. The zero-one laws of Hewitt and Savage [Trans. Amer. Math. Soc. 80 (1955) 470-501] and Horn and Schach [Ann. Math. Statist. 41 (1970) 2130-2131] follow as special cases of our result.


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Julius R. Blum. Pramod K. Pathak. "A Note on the Zero-one Law." Ann. Math. Statist. 43 (3) 1008 - 1009, June, 1972.


Published: June, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0245.60028
MathSciNet: MR300314
Digital Object Identifier: 10.1214/aoms/1177692564

Rights: Copyright © 1972 Institute of Mathematical Statistics


Vol.43 • No. 3 • June, 1972
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