Slow convergences of ergodic averages

Birkhoff’s theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $\psi (n)\to +0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the corresponding time averages converged a.e. with a rate slower than $\psi$. We prove again similar statements answering a question of I. Podvigin in passing.