In this lecture we state and prove the famous Birkhoff Ergodic Theorem.

Suppose $f$ is a ergodic[1] dynamical system on a probability space $(X,\mathscr{A},\mu)$. Given an $L^1$ function $u$, there are two natural ways to define the average of $u$.

• We define the time average of $u$ (with respect to $f$) by $$\widehat{u}(x) := \lim_{k \to \infty} \frac{1}{k}\sum_{i=0}^{k-1} u(f^i (x)).$$ A priori, it is not obvious this limit exists (but it does.)
• We define the space average of $u$ to be its integral: $\int_X u \, d \mu$.

The Birkhoff Ergodic Theorem states that the time average $\widehat{u}$ is constant almost everywhere, and moreover that this constant is equal to the space average. Thus:
$$\textbf{time average = space average.}$$

1. Actually in class we prove a more general result that does not require the system to be ergodic. The version discussed here is the most useful special case though. ↩︎