Almost Everywhere Convergence of Ergodic Averages of Nonlinear Operators

For order preserving mappings T, which are nonexpansive in L1 and L∞, two principal, mutually equivalent results are shown. The almost everywhere convergence of (Tnf/n) (primarily of interest if T0 ≠ 0) and of the ergodic averages Snf/(n + 1)), where S0f = f, Sn + 1f = f + TSnf.