INDIVIDUAL ERGODIC THEOREMS FOR INFINITE MEASURE

Given a sigma-finite infinite measure space (Omega, mu), it is shown that any Dunford-Schwartz operator T : L-1(Omega) - L-1(Omega) can be uniquely extended to the space L-1(Omega) + L-infinity(Omega). This allows one to find the largest subspace R-mu of L-1(Omega) + L-infinity(Omega) such that the ergodic averages n(-1) Sigma(n-1)(k =0) T-k(f) converge almost uniformly (in Egorov's sense) for every f is an element of R-mu and every Dunford-Schwartz operator T. Utilizing this result, almost uniform convergence of the averages n(-1) Sigma(n-1)(k =0) beta T-k(k)(f) for every f is an element of R-mu, any Dunford- Schwartz operator T and any bounded Besicovitch sequence {beta(k)} is established. Further, given a measure preserving transformation tau : Omega - Omega, Assani's extension of Bourgain's Return Times theorem to sigma-finite measures is employed to show that for each f is an element of R-mu there exists a set Omega(f) subset of Omega such that mu(Omega \ Omega(f)) = 0 and the averages n(-1) Sigma(n-1)(k =0) beta(k)f (tau(k)omega) converge for all omega is an element of omega(f) and any bounded Besicovitch sequence {beta(k)}. Applications to fully symmetric subspaces E subset of R-mu are outlined.