invariantmeasure

An invariant measure is a measure μ defined on a measurable space (X, Σ) that remains unchanged under a transformation T: X → X, in the sense that μ(T^{-1}(A)) = μ(A) for all A ∈ Σ. More generally, if a group G acts on X via measurable maps g: X → X, μ is invariant if μ(gA) = μ(A) for all A and all g ∈ G. In dynamical systems, such measures are used to describe statistical properties of orbits, and they are called T-invariant or G-invariant measures.

Examples include Lebesgue measure on R^n, which is invariant under translations; the Haar measure on a locally compact group, which is left-invariant (and unique up to scaling); and the counting measure on a set is invariant under all bijections.

Existence: on compact spaces, Krylov–Bogolyubov theorem guarantees at least one invariant probability measure for any continuous map. Uniqueness occurs in special cases; for a compact group, Haar measure is the unique left-invariant probability measure. In dynamical systems, invariant measures may be finite or infinite.

Ergodicity: an invariant measure is ergodic if every invariant set has measure 0 or 1. Ergodic measures capture indecomposable statistical behavior. The ergodic decomposition theorem states that any invariant probability measure can be represented as an integral of ergodic measures. In Markov chains and stochastic processes, an invariant measure is often called a stationary distribution, satisfying πP = π.