a measure-preserving dynamical system
A measure-preserving dynamical system packages all the data ergodic theory needs into one object: the quadruple (Omega, F, P, T), where (Omega, F, P) is a probability space and T : Omega -> Omega is a measure-preserving transformation. It is the deterministic engine — T — running on a probabilistic stage — P — and ergodic theory is the study of what such an engine does over the long run.
Two systems (Omega, F, P, T) and (Omega', F', P', T') are isomorphic if there is a measure-preserving bijection phi (defined off a null set) with phi compose T = T' compose phi almost everywhere, i.e. a relabelling of points that intertwines the dynamics. Isomorphism is the right notion of "same system", and a central programme of the subject (the isomorphism problem) is to decide when two systems are isomorphic; invariants such as ergodicity, mixing and Kolmogorov-Sinai entropy are precisely the tools built to tell systems apart. One can run T in discrete time (the iterates T^n, n >= 0 or n in Z if invertible) or replace T by a one-parameter flow (T_t) for continuous time; the discrete case is the default here.
Why this abstraction earns its keep: a vast range of objects are special cases. Stationary sequences, equidistribution problems in number theory, geodesic and Hamiltonian flows in physics, and Markov chains in their stationary regime all become (Omega, F, P, T). Proving one theorem about (Omega, F, P, T) — say Birkhoff's — instantly yields a law of large numbers for all of them. The price of the generality is that the abstract space Omega is often huge and featureless; the art is choosing a concrete model (a shift space, an interval map) that makes the dynamics legible.
Take Omega = {0,1}^N (one-sided binary sequences), P the product of fair coin tosses, F the product sigma-algebra, and T the shift (T omega)_n = omega_(n+1). Then (Omega, F, P, T) is the canonical model of an iid fair-coin process: a measure-preserving dynamical system whose Birkhoff averages are the sample frequencies of heads.
The Bernoulli shift — the simplest non-trivial m.p.s., and the dynamical face of the strong law of large numbers.
Almost everything is stated "almost everywhere": null sets are invisible to the measure, so isomorphism, T-invariance and the ergodic theorems all hold mod P-null sets, never pointwise on every single x.