Maximal Ergodic Theorem On Weighted $L_w^p(X)$ spaces

In this paper, we study the maximal ergodic operator on $L_w^p(X,\mathcal{B},\mu )$ spaces, $1\le p<\mathrm{\infty }$, where $(X,\mathcal{B},\mu )$ is a probability space equipped with an invertible measure preserving transformation $U$ and $w$ is an ergodic $A_p$ weight using transference method.