Cesaro bounded.

Question

The exercise is from Ulrich Krengel's book, Ergodic Theorems, on pages 173-174.

First preliminary notions:

1. a function $h$ with $T^*h=h$ is called harmonic, where $T$ is a contraction in $L_1$.

2. $Y= \{ e>0 \}$ where $e$ is harmonic.

Now prove as an exercise that:

If $T$ is Cesaro bounded and $f \in L_1(\mu)$, $A_n f$ converges stochastically in $Y$.

$$A_n f = \frac{1}{n} \sum_{k=0}^{n-1}T^k f$$

And Cesaro bounded means that $||A_n||_{\infty}$ is unifromly bounded