The representation of an invariant measure μ on a measure-preserving system as an integral (mixture) of ergodic invariant measures: μ = ∫ μ_x dν(x), where almost every μ_x is ergodic and ν is a measure on the space of ergodic components.

Principle

Invariant measures are convex combinations of extreme (ergodic) invariant measures; ergodic measures act as atomic building blocks for statistical behavior.

Demonstration

For a shift-invariant probability measure on a symbolic space that decomposes according to ergodic measures supported on minimal subsystems, time averages for an initial distribution equal averages taken over the corresponding ergodic component.

Misapplication

Assuming the decomposition is always finite or atomic, or applying it when no invariant measure is specified; confusing decomposition of measures with decomposition of the underlying space into disjoint invariant sets in all cases.

Consequence

Time averages for almost every initial condition can be described as integrals against an ergodic component, reducing the study of long-term behavior to ergodic measures.

Reversal

A single ergodic measure (pure component) rather than a mixture: no nontrivial decomposition exists when μ is ergodic.

Boundary

Applies only to invariant measures on measurable dynamical systems; uniqueness or explicit form of the decomposition may fail without separability or standard Borel structure.

Semantic Tension

Often contrasted with spectral decompositions of operators (linear spectral theory) which decompose functions rather than measures into orthogonal modes.

Synthesis

A canonical expression of an invariant measure as an integral over ergodic, indecomposable invariant measures that isolates irreducible statistical behaviors.