Definition:Invariant Measure

Definition
Let $\left({X, \Sigma, \mu}\right)$ 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 $\theta$ iff:


 * $\forall E \in \Sigma: \mu \left({\theta^{-1} \left({E}\right) }\right) = \mu \left({E}\right)$

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


 * $\theta_* \mu = \mu$

Also see

 * Translation-Invariant Measure, an example of an invariant measure