Invariant measure

In mathematics, an invariant measure is a measure that is preserved by some function. Invariant measures are of great interest in the study of 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,

:$mu\; left(\; f^\{-1\}\; (A)\; ight)\; =\; mu\; (A).$

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". Explicity, "μ" is invariant if and only if

:$mu\; left(\; varphi\_\{t\}^\{-1\}\; (A)\; ight)\; =\; mu\; (A)\; forall\; t\; in\; T,\; A\; in\; Sigma.$

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".

Examples

* Consider the real line R with its usual Borel σ-algebra; fix "a" ∈ R and consider the translation map "T"_"a" : R → R given by:

::$T\_\{a\}\; (x)\; =\; x\; +\; a.$

: Then one-dimensional Lebesgue measure "λ" is an invariant measure for "T"_"a".

* More generally, on "n"-dimensional Euclidean space R^"n" with its usual Borel σ-algebra, "n"-dimensional Lebesgue measure "λ"^"n" is an invariant measure for any isometry of Euclidean space, i.e. a map "T" : R^"n" → R^"n" that can be written as

::$T(x)\; =\; A\; x\; +\; b$

: for some "n" × "n" orthogonal matrix "A" ∈ O("n") and a vector "b" ∈ R^"n".