the ergodic decomposition
The ergodic decomposition is the structure theorem that says: every stationary (invariant) measure is, in a precise sense, a mixture of ergodic ones. Ergodic systems are the indivisible atoms of measure-preserving dynamics, and the decomposition shows that the non-ergodic case is never genuinely new — it is just an average of ergodic cases. This is why so much of the theory can focus on the ergodic case without loss of generality.
Fix a transformation T (or a flow) on a nice (standard Borel / Polish) space and consider the set of all T-invariant probability measures. This set is convex, and its extreme points are exactly the ergodic measures: an invariant measure is ergodic if and only if it cannot be written as a nontrivial convex combination of two distinct invariant measures. The ergodic decomposition theorem (a form of Choquet's theorem) states that for any invariant measure P there is a probability measure on the set of ergodic measures such that P = integral mu d(weighting), i.e. P = integral P_omega dP(omega) where omega -> P_omega assigns to almost every point the ergodic component containing it. Dynamically, the invariant sigma-algebra I slices Omega into ergodic pieces; conditioning on I, P(. given I)(omega) is itself ergodic, and integrating back over I recovers P. Birkhoff's limit E[ f given I ] is precisely the space-average of f within the ergodic component you happen to be in.
The decomposition reorganises the whole subject. It lets one reduce general theorems to the ergodic case (prove it for each component, integrate), it gives the right picture of Birkhoff's non-constant limit (it is the per-component mean), and it is the engine behind the de Finetti theorem read dynamically (an exchangeable sequence is a mixture of iid sequences — a mixture of ergodic components). The technical hypothesis to respect: the clean theorem needs the space to be standard Borel (Polish), so that regular conditional probabilities exist; on pathological measurable spaces a genuine decomposition can fail. Within that standard setting, though, the picture is exact: study the atoms (ergodic measures), and the rest is averaging.
Toss one biased coin with random bias p ~ Uniform[0,1], then generate an infinite iid Bernoulli(p) sequence. The resulting law is stationary but NOT ergodic; its ergodic decomposition is P = integral_0^1 (Bernoulli(p)^N) dp — each ergodic component is the iid sequence at a fixed p, and the sample frequency converges to that random p (which is E[ X_1 given I ]), never to the overall mean 1/2.
A non-ergodic stationary law as a Choquet integral of its ergodic components — the dynamical face of de Finetti's theorem.
The decomposition requires a standard (Polish) state space so that regular conditional probabilities exist; on a pathological measurable space the ergodic components may not assemble into a genuine measure. Within that setting the components are essentially unique.