Definition:Invariant Measure
Jump to navigation
Jump to search
Definition
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:
- $\theta_* \mu = \mu$
Examples
Arcs on Unit Circle
Let $\CC$ denote the unit circle embedded in the complex plane:
- $\CC = \set {z \in \C: \cmod z = 1}$
The standard measure for arcs on $\CC$ is invariant under the mapping $\map T z = z^2$
Also see
- Definition:Measure-Preserving Transformation
- Definition:Translation Invariant Measure: an example of an invariant measure
- Poincaré Recurrence Theorem: if $\mu$ is a probability measure, then $\theta$ has a recurrence property
- Results about invariant measures can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): invariant measure
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 5$: Problem $9$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): invariant measure