Linear growth bound for integrable observables
= Linear growth bound for integrable observables
In a probability <measure-preserving system>, $f\in L^1$ implies $f(T^nx)/n\to0$ <almost everywhere>. For every $\varepsilon>0$, the <tail integral formula for moments> gives $\sum_{n\geq1}\mu(|f|>\varepsilon n)\leq\|f\|_1/\varepsilon$. Invariance and the <Borel-Cantelli lemma> finish the proof. Consequently division by $n^a$ also gives zero for every $a\geq1$.