On the Individual Ergodic Theorem for $K$-Automorphisms

Abstract

Let $(X, \mathscr{B}(X), P)$ be a probability space and let $T$ be a $K$-automorphism. If $T$ satisfies a Rosenblatt mixing condition of a certain kind, we show that if $\{k_n\}^\infty_{n=1}$ is an arbitrary increasing sequence of integers and $g$ belongs to a certain class of functions then $$\lim_{n\rightarrow\infty} \frac{1}{n} \sum^n_{j=1} g(T^{k_j}x) = E(g) \mathrm{a.s.}$$