Let $(X, \borel, \mu)$ be a probability space, $f \in L^1(\mu)$ and $T\colon X \to X$ a measure preserving transformation. Birkhoff's ergodic theorem (often called the pointwise or strong ergodic theorem) states that there exists $f^*\in L^1(\mu)$ such that \begin{equation*} \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1} f(T^k x) = f^*(x) \end{equation*}for almost all $x\in X$ Moreover, $f^*$ is $T$ invariant (i.e., $f^*\circ T = f^*$ almost everywhere and $$\int f^*d\mu = \int f d\mu.$$ In particular, if $T$ is ergodic then the $T$ invariance of $f^*$ implies that it is constant almost everywhere, and so this constant must be the integral of $f^*$ that is, if $T$ is ergodic, then $$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1} f(T^k x) = \int fd\mu$$ for almost every $x$ This is often interpreted in the following way: for an ergodic transformation, the time average equals the space average almost surely.