Maximal ergodic theorem

Statement

Let $T$ preserve a probability measure and let $f\in L^1(X,\mu;\mathbb R)$ . Set

S_nf=\sum_{j=0}^{n-1}f\circ T^j,\qquad E_N=\left\{x:\max_{1\leq n\leq N}S_nf(x)>0\right\}.

The maximal ergodic theorem gives $\int_{E_N} f\,d\mu\geq0$ . The same conclusion holds for $E=\{\sup_{n\geq1}S_nf>0\}$ .

Proof of the finite statement

Put $M_N=\max(0,S_1f,\ldots,S_Nf)$ . On $E_N$ , separating the first summand gives $M_N\leq f+M_N\circ T$ . Since $M_N=0$ off $E_N$ and $M_N\geq0$ , measure preservation gives

\int_{E_N}f\,d\mu\geq\int M_N\,d\mu-\int_{E_N}M_N\circ T\,d\mu\geq0.

The infinite statement follows from dominated convergence as $E_N$ increases to $E$ .

Maximal inequality

Apply the theorem to $|h|-a$ , for $h\in L^1$ and $a>0$ . With the time averages $A_n$ , it gives

\mu\{\sup_{n\geq1} A_n|h|>a\}\leq\frac{\|h\|_1}{a}.

This controls the set of trajectories with a large averaging error. It is the key estimate for extending pointwise convergence from well-behaved observables to all of $L^1$ .

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.5, maximal ergodic inequality.