Invariant measure

In mathematics, an invariant measure is a measure that is preserved by some function. Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.

Let (X, Σ) be a measurable space and let f be a measurable function from X to itself. A measure μ on (X, Σ) is said to be invariant under f if, for every measurable set A in Σ,

In terms of the push forward, this states that f_∗(μ) = μ.

The collection of measures (usually probability measures) on X that are invariant under f is sometimes denoted M_f(X). The collection of ergodic measures, E_f(X), is a subset of M_f(X). Moreover, any convex combination of two invariant measures is also invariant, so M_f(X) is a convex set; E_f(X) consists precisely of the extreme points of M_f(X).

In the case of a dynamical system (X, T, φ), where (X, Σ) is a measurable space as before, T is a monoid and φ : T × X → X is the flow map, a measure μ on (X, Σ) is said to be an invariant measure if it is an invariant measure for each map φ_t : X → X. Explicitly, μ is invariant if and only if

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