Nonnegative ergodic averages with infinite integral ($A_{N}f\to +\infty$)

In a probability measure-preserving system whose transformation is an ergodic transformation, a nonnegative measurable $f$ with $\int fd\mu =+\infty$ has $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to +\infty$ almost everywhere. Apply the pointwise ergodic theorem to $\min (f,K)$ on a common full-measure set for integer $K$, then bound the lower limit by each truncated integral. The monotone convergence theorem sends those integrals to infinity. The finite-integral case is the ordinary pointwise ergodic theorem.