Definition:Measure-Preserving Transformation

Definition

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

Let $T: X \to X$ be a measurable mapping.

$T$ is called a measure-preserving transformation if and only if $\mu$ is invariant under $T$.

Also known as

More explicitly, $T$ is also called a $\mu$-preserving transformation.

Also see

• Definition:Invariant Measure
• Definition:Measure-Preserving Mapping
• Definition:Ergodic Measure-Preserving Transformation
• Definition:Measure-Preserving Dynamical System

Sources

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