On the Existence of Invariant Measures

Let P b e a measurable transformation on a measure space (Q, Cfc, P ) , with 0<P(£2) < oo. Call T absolutely continuous if P(A) = 0 implies P(T~*A)=0. The transformation T is said to have the Birkhoff recurrence property if, for each -4EG*, limn-co(lA0 2"~o XA{T 'CO) exists for almost all co£0. I t has been shown that if T is absolutely continuous and has the Birkhoff recurrence property, then there exists a non-negative, finite, countably additive measure Q on & such that (i) Q « P , (ii) Q and P agree on invariant sets, (iii) Q(A) = Q(T~A) for each 4 £ û [3 ]. In this paper we prove the following result.