Definition:Invariant Measure

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\theta: X \to X$ be an $\Sigma / \Sigma$-measurable mapping.

Then $\mu$ is said to be a $\theta$-invariant measure or to be invariant under $\theta$ :


 * $\forall E \in \Sigma: \map \mu {\theta^{-1} \sqbrk E} = \map \mu E$

where $\theta^{-1} \sqbrk E$ denotes the preimage of $E$ under $\theta$.

In terms of a pushforward measure, this can be concisely formulated as:


 * $\theta_* \mu = \mu$

Also known as
Equivalently, $\theta$ is said to be $\mu$-preserving transformation.

Also see

 * Translation-Invariant Measure, an example of an invariant measure
 * Poincaré Recurrence Theorem: if $\mu$ is a probability measure, then $\theta$ has a recurrence property