a measure-preserving transformation

Ergodic theory studies dynamics that respect a probability measure: a map T from a probability space (Omega, F, P) to itself that pushes the measure forward to itself, so that the statistical picture looks the same after you apply T. This is the abstract setting in which the question "do time averages equal space averages?" is even well posed, and it is the natural home for any stationary stochastic process.

Precisely, T : Omega -> Omega is measure-preserving (m.p.) if it is measurable and P(T^(-1) A) = P(A) for every A in F. The pullback T^(-1) is used, not the forward image, because that is what behaves well for non-invertible maps and for integration: a clean equivalent statement is that E[f compose T] = E[f] for every integrable f, i.e. integrating f "one step later" gives the same answer. Iterating, T^n is measure-preserving for all n, and the orbit Omega -> T Omega -> T^2 Omega ... is a deterministic flow that nonetheless preserves all probabilities. When T is invertible with measurable inverse and both T and T^(-1) are m.p., one speaks of an automorphism.

Measure preservation is exactly the hypothesis that makes the ergodic theorems run; drop it and Birkhoff's averages need not converge to anything sensible. It is also exactly the abstract content of stationarity: a real-valued process (X_0, X_1, ...) is stationary if and only if it is generated by a coordinate function under the shift on a measure-preserving system. So "measure-preserving transformation" and "stationary process" are two languages for one idea.