Birkhoff’s pointwise ergodic theorem

For an integrable real or complex observable, average its values at the first N nonnegative iterates, starting at time zero. Birkhoff's theorem says these averages converge almost everywhere and in L1 on a probability-preserving system. No inverse transformation or independence assumption is needed for this forward theorem. Integrability is essential to this stated version: merely being measurable does not control rare large observations. The exceptional null set can depend on the observable, so convergence for each integrable function is not automatically a simultaneous assertion about all functions on every point.

The limiting function is invariant up to null sets. One intuition is that shifting the starting position changes only the first and last terms of a long average. Turning that intuition into a pointwise proof for an unbounded integrable function requires control of those boundary terms and a maximal argument; it is not justified simply by saying two terms divided by N vanish. We invoke the full pointwise theorem and use exact periodic examples to inspect what its invariant limit means.

An event is invariant modulo null sets when pulling it back by T changes it only on a null set. These events form the invariant sigma-algebra. Its measurable observables cannot change along typical trajectories, so averaging cannot remove their component information. The theorem identifies the time-average limit by integrating it against every invariant event: its integral there equals the original observable's integral. These are exactly the defining identities for conditional expectation on the invariant sigma-algebra.

Ergodicity means every invariant event has probability zero or one. Then every integrable invariant function is almost surely constant, and its preserved integral forces that constant to be the global mean. If ergodicity fails, different invariant regions can have different means, even when all orbits are perfectly periodic and every average converges. A global distribution over starting states mixes these regions; a single trajectory retains the information specifying its own region and generally cannot reproduce the mixture's average.

For the indicator of an event, the time average is its empirical visit frequency. In an ergodic probability system, that frequency equals the event's probability for almost every starting state. In a nonergodic system it equals the invariant conditional probability of the event, which can vary across components. This is much stronger than infinitely many returns, but still gives no general finite-N convergence rate. Very slow averages are compatible with the qualitative theorem.

The finite-probability setting must remain explicit. A finite nonzero invariant measure can be normalized before applying the theorem, while an infinite measure cannot be normalized this way. In infinite-measure dynamics one needs other formulations and sometimes different normalizations, rather than a conditional probability expectation copied from the probability case. Similarly, pointwise convergence of orbit averages should not be confused with convergence of distributions or with decorrelation: an ergodic periodic cycle has correct time averages and nevertheless retains periodic correlations forever.