= Nonnegative ergodic averages with infinite integral
{title2=$A_Nf\to+\infty$}
In a probability <measure-preserving system> whose transformation is an <ergodic transformation>, a nonnegative measurable $f$ with $\int f\,d\mu=+\infty$ has $N^{-1}\sum_{j=0}^{N-1}f(T^jx)\to+\infty$ <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>.
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