Let $E\subseteq (0,1)$ be a meaurable subset with Lebesgue measure $\lambda (E)$. Is it true that, for almost all $\alpha$,$$\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{1\le k\le n}{1}_{\{k\alpha \}\in E}=\lambda (E)$$for all $E$?