Definition:Invariant Measure

Definition
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

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

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


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

Also see

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