On Ergodic Averages and Absorbing Sets for Positive Contractions in L1

Let T: L1(X) → L1(X) be a positive contraction, that is, a linear operator satisfying ||T|| ≤ 1 and T(L+1(X)) ⊆ L+1(X). Let Sn = Σn−1k=0Tk, An = (1/n) Sn, and let F = {x: limsupn→∞Anφ(x) = ∞}, where φ is a.e. positive. First, we prove that for every sequence (ƒn) in L+1(X) such that, for every m, (Tm ƒn − ƒn)+ tends stochastically to 0 on F, min (ƒn, 1/ƒn) tends stochastically to 0 on F. This implies, in particular, the Stochastic Ergodic Theorem. Second, we study the behavior of sequences of the form anSk(n) φ(x) to obtain decompositions of the conservative part of X into six absorbing sets