Ergodic theory
Dynamical systems
The study of the long-run statistical behaviour of measure-preserving dynamical systems, where averages along a single trajectory can be compared with averages over the whole space.
What is usually left out
A system is ergodic when those two averages agree for almost every starting point — time spent in a region matches that region’s size. This is what licenses replacing an intractable average over all states with the observation of one long trajectory, and it fails for systems that decompose into separate invariant pieces.