L1 convergence in the Birkhoff ergodic theorem on a finite measure space
= L1 convergence in the Birkhoff ergodic theorem on a finite measure space
{title2=$A_nf\to f^*$ in $L^1$}
On a <finite measure> space, the convergence in the <Birkhoff ergodic theorem> also holds in $L^1$. Prove it first for bounded truncations by the <dominated convergence theorem>, then use the $L^1$ contraction $\|A_nh\|_1\leq\|h\|_1$ and <Fatou lemma> to remove the truncation.