L1 Mean Ergodic Theorem

Theorem

Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.

Let $\map {L^1_\C} \mu$ be the complex-valued $L^1$ space of $\mu$.

Then for each $f \in \map {L^1_\C} \mu$ there is a $T$-invariant function $f' \in \map {L^1_\C} \mu$ such that:

$\ds \lim_{N \mathop \to \infty} \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f \circ T^n = f'$ converges in the $L^1$-norm

Proof

For $c > 0$ let:

$f_c = f \cdot \chi_{\set {\cmod f \le c} }$

where $\chi_B$ is the characteristic function of $B$.

Then $f_c \in \map {L^2_\C} \mu$ and:

\(\ds f - f_c\)

\(=\)

\(\ds f \cdot \paren {1 -\chi_{\set {\cmod f \le c} } }\)

\(\ds \)

\(=\)

\(\ds f \cdot \chi_{\set {\cmod f > c} }\)

\(\ds \)

\(\to\)

\(\ds 0\)

in $\map {L^1_\C} \mu$ as $c \to +\infty$

Let $\epsilon > 0$.

Let $c > 0$ be such that:

$\norm {f - f_c}_1 \le \frac \epsilon 3$

As $\norm \cdot_1 \le \norm \cdot_2$, by Mean Ergodic Theorem there exists an $N_0 \in \N_{>0}$ such that:

$\ds \forall M,N \ge N_0 : \norm { \dfrac 1 M \sum_{n \mathop = 0}^{M - 1} f_c \circ T^n - \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f_c \circ T^n}_1 \le \frac \epsilon 3$

Thus:

$\ds \forall M,N \ge N_0 : \norm { \dfrac 1 M \sum_{n \mathop = 0}^{M - 1} f \circ T^n - \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f \circ T^n}_1 \le \epsilon$

By Riesz-Fischer Theorem there exists an $\tilde f \in \map {L^1_\C} \mu$ such that:

$\ds \lim_{N \to \infty} \norm { \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f \circ T^n - \tilde f}_1 = 0$

In addition:

\(\ds \norm {\tilde f \circ T - \tilde f}_1\)

\(=\)

\(\ds \lim_{N \to \infty} \norm {\dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f \circ T^{n+1} - \dfrac 1 N \sum_{n \mathop = 0}^{N - 1} f \circ T^n}_1\)

\(\ds \)

\(=\)

\(\ds \lim_{N \to \infty} \norm {\dfrac {f \circ T^N - f} N}_1\)

\(\ds \)

\(\le\)

\(\ds \lim_{N \to \infty} \dfrac {\norm {f \circ T^N}_1 + \norm f_1} N\)

\(\ds \)

\(=\)

\(\ds \lim_{N \to \infty} \dfrac {2 \norm f_1} N\)

\(\ds \)

\(=\)

\(\ds 0\)

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Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory $2.5$ The Mean Ergodic Theorem