Abstract
Let ℳ denote the class of infinite product probability measures μ=μ1×μ2×⋯ defined on an infinite product of replications of a given measurable space (X, A), and let ℋ denote the subset of ℳ for which μ(A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member μ, of ℳ belongs to ℋ. In the present paper, the class ℋ is shown to possess several closure properties. E.g., if μ∈ℋ and μ0≪μn for some n ≧1, then μ0×μ1×μ2×...∈ℋ. While the current results do not permit a complete characterization of ℋ they demonstrate conclusively that ℋ is a much larger subset of ℳ than previous results indicated. The interesting special case X={0,1} is discussed in detail.
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