Invariant measure

Remarks

Equivalently, the pushforward measure $T_* \mu$ equals $\mu$, or $T$ is a measure-preserving transformation of $(X, \mathcal{A}, \mu)$ to itself. For an integrable function $g$, invariance gives $\int g \circ T \, d\mu = \int g \, d\mu$. The invariant probability measures of $T$ form a convex set, whose extreme points are the ergodic ones.

Example: the doubling map $T(x) = 2x \bmod 1$ on $[0, 1)$ and Lebesgue measure $\lambda$. For an interval $[a, b) \subseteq [0, 1)$, one has $T^{-1}([a, b)) = [a/2, b/2) \cup [(a+1)/2, (b+1)/2)$, of length $b - a$. Since intervals generate the Borel sets and are closed under intersection, $\lambda$ is $T$-invariant.

Non-example: the map $T(x) = x^2$ on $[0, 1]$ with Lebesgue measure $\lambda$. One has $T^{-1}([0, 1/4]) = [0, 1/2]$, so $\lambda(T^{-1}([0, 1/4])) = 1/2 \ne 1/4$, and $\lambda$ is not $T$-invariant. The point masses at $0$ and at $1$ are $T$-invariant, since $0$ and $1$ are fixed points.