Ergodicity is a concept from statistical mechanics and dynamical systems theory that describes the behavior of systems over time. In general terms, a system is considered ergodic if its time averages are equivalent to its ensemble averages. This means that a sufficiently long observation of a single trajectory (or the time evolution of a single state of the system) will provide the same statistical properties as observing a large number of different states of the system at a single point in time (the ensemble).
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A measure-preserving transformation is ergodic when every set in its invariant sigma-algebra has measure zero or full measure. Equivalently, every integrable invariant function is almost everywhere constant.