A Note on the Proof of the Zero-One Law of Blum and Pathak

Abstract

Let $\{(\Omega, \mathscr{A}, \mu_n), n \geqq 1\}$ be a sequence of probability spaces. Blum and Pathak [Ann. Math. Statist. 43 (1972) 1008-1009] proved a zero-one law for permutation invariant sets $A \epsilon \mathscr{A}^\infty$; which includes 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] as special cases. The proper reason for this is shown to be the fact that the set of measures admitting the zero-one law of Blum and Pathak coincides with the set of all strong limit points of measures admitting the zero-one law of Horn and Schach.