TheoremProved

Birkhoff ergodic theorem

Statement

Let $(X,\mathcal{B},\mu,T)$ be a measure-preserving system with $\mu$ a probability measure, and $f \in L^1(\mu)$ . Then the time averages $\frac{1}{n}\sum_{k=0}^{n-1} f(T^k x)$ converge $\mu$ -almost everywhere to a $T$ -invariant function $f^*$ with $\int f^* \,d\mu = \int f\,d\mu$ ; if $T$ is ergodic, $f^* = \int f\,d\mu$ almost everywhere.

Why is it true?

Watching a single trajectory for a very long time and averaging what you see along the way gives, almost always, the same answer as averaging over the whole space at one instant — provided the system does not break into pieces the trajectory cannot mix between. Time spent equals space explored, on average, once the system is ergodic.

Proof sketch

The standard proof uses the maximal ergodic theorem: for $f \in L^1$ and $Mf = \sup_{n \ge 1} \frac{1}{n}\sum_{k=0}^{n-1} f\circ T^k$ , one shows $\int_{\{Mf > 0\}} f\, d\mu \ge 0$ via a telescoping/maximal-function argument. Applying this to $f - a$ and $b - f$ for rationals $a > b$ on the $T$ -invariant set where the $\limsup$ of the time averages exceeds $a$ and the $\liminf$ is below $b$ shows that this set has measure zero, forcing almost-everywhere convergence to a limit function $f^*$ ; invariance of $f^*$ follows since shifting by $T$ does not change the time average in the limit, and the identity of integrals follows from the dominated convergence theorem applied along a truncation argument.

Topics that use this theorem

Ergodic theory

Related theorems

Strong law of large numbers

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Peter Walters (1982). An Introduction to Ergodic Theory