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.
Definition.
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
Put another way, "μ" is an invariant measure for a sequence of random variables ("Z""t")"t"≥0 (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition "Z"0 is distributed according to "μ", so is "Z""t" for any later time "t".
When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of 1, this being the largest eigenvalue as given by the Frobenius-Perron theorem.