In the Birkhoff ergodic theorem we have a PMPS $(X,B,\mu,T)$ and that for any $f\in L^1(X,\mu)$ $\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)\to \int f d\mu,$ in measure, in $L^1$-norm and $\mu$-a.e. My question is: what is, given $\epsilon>0,$ the estimation of $\mu\left(x:\left|\frac{1}{N}\sum_{n=0}^{N-1}f(T^n x)-\int fd\mu\right|>\epsilon\right)$ when $N\in \mathbb{N}$ is big? I am looking for a proof or reference that I could not yet not found.