Definition:Invariant Measure

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

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

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

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

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 see

  • Results about invariant measures can be found here.