Let $(\Omega, \cf, \bp, T)$ be a [[Measure System|measure system]], then the following are equivalent: 1. $(\Omega, \cf, \bp, T)$ is [[Ergodic Measure System|ergodic]]. 2. For all $X \in L^1$, $\limv{n}\frac{1}{n}\sum_{k \le n}X \circ T^k = \ev(X)$ ([[Almost Everywhere|a.s.]]). 3. For all $X \in L^\infty$, $\limv{n}\frac{1}{n}\sum_{k \le n}X \circ T^k = \ev(X)$ ([[Almost Everywhere|a.s.]]). 4. For all $E \in \cf$, $\limv{n}\frac{|\bracs{k \le n: T^kx \in E}|}{n} = \bp\bracs{U}$ (a.s.). *Proof*. Suppose that $T$ is ergodic. Let $X \in L^1$ and assume without loss of generality that $\ev(X) = 0$. First note that $\ev\braks{X \circ T^n} = \frac{1}{n}\sum_{1 \le k \le n}\ev[X \circ T^k]$ for all $n \in \nat$. Let $f^* = \limsup_{n \to \infty}\frac{1}{n}\sum_{k \le n}X \circ T^k$, then $f^*$ is $T$-invariant and constant a.s. For almost every $\omega \in \Omega$, there exists $n \in \nat$ such that $\frac{1}{n}\sum_{k \le n}X(x) \circ T^k \ge f^*/2$. Let $ E_n = \bracs{\omega \in \Omega: \frac{1}{n}\sum_{k \le n}X(x) \circ T^k \ge f^*/2} $ then $\bp\bracs{E_n} \upto 1$ as $n \to \infty$. $n_x$ be the minimum of such $n$s. Let $f$ be a measurable function and $\eps > 0$,