Birkhoff's ergodic theorem
/ BIRK-hoff /
Birkhoff's theorem is the central theorem of ergodic theory and the deepest generalisation of the strong law of large numbers: it says that for a measure-preserving system the time average of an observable along almost every orbit converges, and identifies the limit. It is the precise sense in which "the average over time exists".
Statement: let (Omega, F, P, T) be measure-preserving and f in L^1(P). Then the time averages A_n f (x) = (1/n) sum_(k=0)^(n-1) f(T^k x) converge for P-almost every x, and the limit equals the conditional expectation E[ f given I ], where I is the invariant sigma-algebra. The limit is also the L^1 limit. If T is ergodic then I is trivial and the limit is the constant E[f] — time average equals space average. The pointwise (almost-sure) convergence is what makes this hard; the modern proof rests on the maximal ergodic theorem (or Garsia's slick maximal inequality), which controls how large the partial averages can get.
Birkhoff's theorem is the master template: feeding in the right (Omega, F, P, T) recovers the strong law for iid sequences (the Bernoulli shift), equidistribution of irrational-rotation orbits (Weyl), the convergence of empirical frequencies of any stationary process, and continued-fraction statistics (the Gauss map). The hypotheses are exactly the two you would guess and neither can be dropped silently: f must be integrable (E|f| < infinity) for the limit to exist, and T must be measure-preserving for the averages to converge at all; ergodicity is an extra hypothesis needed only to force the limit to be constant. The honest caveat: without ergodicity the limit is a genuine random variable, taking different values on different ergodic components, so "converges" does not mean "converges to the mean".
On [0,1) with the Gauss map T(x) = 1/x mod 1 and its invariant Gauss measure dP = (1/log 2) dx/(1+x), Birkhoff applied to f = 1_{first continued-fraction digit = 1} gives that, for Lebesgue-almost every x, the asymptotic frequency of the digit 1 in its continued fraction is log_2(4/3) — a space average, computed from the invariant measure.
Birkhoff turns a question about almost every individual orbit into a single integral against the invariant measure.
Convergence of (1/n) sum f(T^k x) is a deep almost-sure statement, not the easy L^2 fact (that is von Neumann's mean ergodic theorem). The maximal ergodic theorem is the engine that upgrades L^2 convergence to almost-everywhere convergence.