\begin{definition}[Definition:Invariant Measure]
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:
:$\theta_* \mu = \mu$
\end{definition}