the von Neumann mean ergodic theorem
/ fon NOY-mahn /
Von Neumann's theorem is the Hilbert-space, L^2 version of the ergodic theorem, and it actually came first (1931, a year before Birkhoff). Where Birkhoff asserts almost-sure convergence of time averages, von Neumann asserts convergence in mean square — a softer mode that is far easier to prove because it is pure operator theory, yet it already pins down the limit.
The setup abstracts the dynamics into an operator. To T associate the Koopman operator U on L^2(P) defined by Uf = f compose T; because T is measure-preserving, U is an isometry (||Uf|| = ||f||), and U is unitary when T is invertible. The theorem says: for any contraction (or isometry) U on a Hilbert space H, the Cesaro averages (1/n) sum_(k=0)^(n-1) U^k f converge in norm to P_inv f, the orthogonal projection of f onto the subspace of U-invariant vectors {g : Ug = g}. The proof is elegant: H splits as the invariant vectors plus the closure of the coboundaries {g - Ug}; averages fix the first part and kill the second by telescoping. Translated back, (1/n) sum f(T^k .) -> E[ f given I ] in L^2, the same limit as Birkhoff but in mean square.
Its value is partly conceptual — it reveals ergodic averaging as a projection onto invariants, a viewpoint that powers spectral theory of dynamical systems (mixing, weak mixing and entropy all get spectral characterisations through U). It is also the technically convenient half of the pair: many results only need L^2 convergence, and von Neumann's theorem extends cleanly to families of operators and to amenable group actions. The honest distinction to keep: mean (L^2) convergence does not by itself give almost-sure convergence — that genuinely stronger conclusion is Birkhoff's, and requires the maximal inequality, not just orthogonality.
For the Koopman operator U of a measure-preserving T, take f with E[f] = 0 on an ergodic system. The invariant subspace is just the constants, so P_inv f = 0, and von Neumann gives || (1/n) sum_(k<n) f compose T^k ||_2 -> 0 — the centred time averages shrink to zero in L^2, a clean ergodic-mean statement proved by orthogonality alone.
Ergodic averaging seen as orthogonal projection onto invariants — the operator-theoretic heart of the subject.
Mean ergodic convergence holds for any contraction on a Hilbert space, with no measure-preservation needed; the dynamical content (and almost-sure convergence) is the extra mile Birkhoff travels.