Some Extensions of the Hewitt-savage Zero-one Law So~ffi Extensions of the Hewitt-savage Zero-one Law

Let M denote the class of infinite product probability l:leaSUres J-1=J-1 1 xJ-l 2 x • • • defined on an infinite product of replications of a given measurable space (X~A), and let H denote the subset of fl for which J-1(A)=O or 1 for each permutation invariant event A. Previous works (referenced in the paper) discuss very restrictive sufficient conditions under which a given member J-I of M belongs to H. In the present paper, the class H is shown to possess several closure properties. E. g., if J-1df and l-I O < < lln for some n;o;l. then lloxJ-li X J-l 2 X • • • eH. Nhile the current results do not permit a complete characterization of H, they demonstrate conclusively that Ii is a r.lUch larger subset of Mthan previous results indicated. The interesting special case X={O,l} is studied in detail.