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.

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


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