Un Theoreme Ergodique Presque Sous-Additif

Abstract

The two following results are proved. Given $(\Omega, \mathscr{F}, \mu, T)$ where $T$ is a measurable transformation preserving the probability measure $\mu$, given a sequence $f_n$ of integrable functions such that $\int (f_{n+k} - f_n - T^nf_k)^+ d\mu \leq c_k \text{with} \lim_k \frac{1}{k} c_k = 0,$ then $(1/n) f_n$ is converging in $L^1$-norm. If, furthermore, $f_{n+k} - f_n - T^nf_k \leq T^nh_k$ with $h_k$ a sequence of positive functions whose integrals are bounded, then $(1/n) f_n$ is also converging a.e. From this extension of Kingman's subadditive ergodic theorem, the Shannon-McMillan-Breiman theorem follows at once.