  The mathematical discipline of ergodic theory developed out of these early ideas. When can a phase average be identified with a time average over infinite time? G. Birkhoff (with earlier results by J. von Neumann) showed that this would be so for all but perhaps a set of measure zero of the trajectories (in the standard measure used to define the probability function) if the set of phase points was metrically indecomposable, that is if it could not be divided into more than one piece such that each piece had measure greater than zero and such that a system started in one piece always evolved to a system in that piece.