Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 4 / 25J / d

Since the measure is finite and $f$ is bounded, $f\in L^1$. By Birkhoff ergodic theorem, $A_{n}f\to f^{\ast }$ almost everywhere. If $|f|\le M$, then $|A_{n}f|\le M$ and therefore $|f^{\ast }|\le M$ almost everywhere. For any finite $p\ge 1$,and this constant is integrable because $\mu (E)<\infty$. The dominated convergence theorem gives $\int |A_{n}f-f^{\ast }{|}^{p}d\mu \to 0$. Thus the ergodic averages converge in every $L^p$, $1\le p<\infty$, without assuming ergodicity.