= Solution
Since the measure is finite and $f$ is bounded, $f\in L^1$. By <Birkhoff ergodic theorem>, $A_nf\to f^*$ almost everywhere. If $|f|\leq M$, then $|A_nf|\leq M$ and therefore $|f^*|\leq M$ almost everywhere. For any finite $p\geq1$,
$$
|A_nf-f^*|^p\leq(2M)^p,
$$
and this constant is integrable because $\mu(E)<\infty$. The <dominated convergence theorem> gives $\int|A_nf-f^*|^p\,d\mu\to0$. Thus \b[the ergodic averages converge in every $L^p$, $1\leq p<\infty$], without assuming ergodicity.
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