Maximal ergodic lemma (${\int }_{E_N}fd\mu \ge 0$)

For a measure-preserving transformation and real integrable $f$, let $S_{k}f={\sum }_{j<k}f\circ T^j$. For $E_N=\{{\max }_{1\le k\le N}{S}_{k}f>0\}$ the lemma asserts ${\int }_{E_N}fd\mu \ge 0$. Passing to the increasing union gives the corresponding infinite-supremum inequality.