The left-hand side of (1) just says how often the orbit of
(that is, the points , , , ...) lies in , and the right-hand side is just the measure
Thus, for an ergodic endomorphism, "space-averages
= time-averages almost everywhere." Moreover, if is continuous and uniquely ergodic with Borel
is continuous, then we can replace the almost everywhere
convergence in (1) with "everywhere."