Ergodic measure

In mathematics, specifically in ergodic theory, an ergodic measure is a measure that satisfies the ergodic hypothesis for a given map of a measurable space into itself. Intuitively, an ergodic measure is one with respect to which the points of the space are "well mixed up" by the map.

Definition

Let ("X", Σ) be a measurable space, let "T" : "X" → "X" be a measurable function and let "f" : "X" → "X" be a measurable and integrable function. A finite measure "μ" : Σ → [0, + ∞] is called an ergodic measure (for the transformation "T" and function "f") if

:$lim\_\{n\; o\; infty\}\; frac\{1\}\{n\}\; sum\_\{k\; =\; 0\}^\{n\; -\; 1\}\; f\; left(\; T^\{k\}\; (x)\; ight)\; =\; frac\{1\}\{mu\; (X)\}\; int\_\{X\}\; f(y)\; ,\; mathrm\{d\}\; mu\; (y)$

for "μ"-almost all points "x" ∈ "X". The quantity on the left-hand side is known as the "(long-)time average" of "f", since it is the average value of "f" over the first "n" iterates of "x" under "T", as "n" tends to infinity; the quantity on the right-hand side is known as the "space average" of "f", since it is simply the average value of "f" over the phase space "X" with respect to the measure "μ". Thus for an ergodic measure, "space average equals time average".

It is conventional, since "μ"("X") is finite, to normalize an ergodic measure so that it is a probability measure, i.e. "μ"("X") = 1.

Properties

Ergodic measures are closely related to invariant measures. It is not hard to show that the collection of invariant probability measures for a given map form a convex subset of the set of all probability measures on the space "X". What is not so obvious is that the ergodic probability measures are precisely the extremal points of the set of invariant probability measures.