Definition:Measure-Preserving Mapping

Definition

Let $\struct {X, \BB, \mu}$ and $\struct {Y, \CC, \nu}$ be probability spaces.

Let $\phi: X \to Y$ be a $\BB / \CC$-measurable mapping.

$\phi$ is said to be measure-preserving if:

$\forall C \in \CC: \map \mu {\phi^{-1} \sqbrk C} = \map \nu C$

Sources

• 2011: Manfred Einsiedler and Thomas Ward: Ergodic Theory: with a view towards Number Theory ... (next) $2.1$: Measure-Preserving Transformations