The Ergodic Decomposition Theorem states any invariant measure decomposes uniquely as an average of ergodic components. The Pinsker factor π(T) is the largest factor with zero Kolmogorov-Sinai entropy — it captures the "deterministic" part. The remaining factor is a Bernoulli shift (Ornstein 1970). KS entropy: h(T) = sup_P h(T,P) = lim H(Pₙ)/n where Pₙ = P ∨ T⁻¹P ∨ ... ∨ T⁻ⁿP. Irrational rotations have h=0 (Pinsker factor = the whole system); doubling map has h=log 2 (fully Bernoulli).