# Birkhoff ergodic theorem

One of the most important theorems in ergodic theory. For an endomorphism $T$ of a $\sigma$-finite measure space $(X,\Sigma,\mu)$, Birkhoff’s ergodic theorem states that for any function $f \in {L^{1}}(X,\Sigma,\mu)$, the limit $$\overline{f}(x) \stackrel{\text{df}}{=} \lim_{n \to \infty} \frac{1}{n} \sum_{k = 0}^{n - 1} f \! \left( {T^{k}}(x) \right)$$ (the time average or the average along a trajectory) exists almost everywhere (for almost all $x \in X$). Moreover, $\overline{f} \in {L^{1}}(X,\Sigma,\mu)$, and if $\mu(X) < \infty$, then $$\int_{X} f ~ \mathrm{d}{\mu} = \int_{X} \overline{f} ~ \mathrm{d}{\mu}.$$
For a measurable flow $(T_{t})_{t \geq 0}$ in a $\sigma$-finite measure space $(X,\Sigma,\mu)$, Birkhoff’s ergodic theorem states that for any function $f \in {L^{1}}(X,\Sigma,\mu)$, the limit $$\overline{f}(x) \stackrel{\text{df}}{=} \lim_{t \to \infty} \frac{1}{t} \int_{0}^{t} f({T_{t}}(x)) ~ \mathrm{d}{t}$$ exists almost everywhere, with the same properties as $f$.