Definition

Definition - Let $(X_1, \mathfrak{B}_1, \mu_1)$ and $(X_2, \mathfrak{B}_2, \mu_2)$ be measure spaces, and $T:X_1 \to X_2$ be a measurable transformation. The transformation $T$ is said to be measure-preserving if for all $A \in \mathfrak{B}_2$ we have that

\begin{equation*} \mu_1(T^{-1}(A)) = \mu_2(A), \end{equation*}where $T^{-1}(A)$ is, as usual, the set of points $x\in X_1$ such that $T(x)\in A$ .

Additional Notation:

• If $T$ is bijective, measure-preserving, and its inverse $T^{-1}$ is also measure-preserving, then $T$ is said to be an invertible measure-preserving transformation.

• Measure-preserving transformations between the same measure space are sometimes called endomorphisms of the measure space.

Remarks:

• The fact that a map $T:X_1 \longrightarrow X_2$ is measure-preserving depends heavily on the sigma-algebras $\mathfrak{B}_i$ and measures $\mu_i$ involved. If other measures or sigma-algebras are also in consideration, one should make clear to which measure space is $T:X_1 \longrightarrow X_2$ measure-preserving.

• Measure-preserving maps are the morphisms on the category whose objects are measure spaces. This should be clear from the next results and examples.

Properties

• The composition of measure-preserving maps is again measure-preserving. Of course, we are supposing that the domains and codomains of the maps are such that the composition is possible.

• Let $(X_1, \mathfrak{B}_1, \mu_1)$ and $(X_2, \mathfrak{B}_2, \mu_2)$ be measure spaces and $(X_1, \overline{\mathfrak{B}_1}, \overline{\mu_1})$ and $(X_2, \overline{\mathfrak{B}_2}, \overline{\mu_2})$ their completions. If $T:(X_1, \mathfrak{B}_1, \mu_1) \longrightarrow (X_2, \mathfrak{B}_2, \mu_2)$ is measure-preserving, then so is $T:(X_1, \overline{\mathfrak{B}_1}, \overline{\mu_1}) \longrightarrow (X_2, \overline{\mathfrak{B}_2}, \overline{\mu_2})$ .

• Let $(X_1, \mathfrak{B}_1, \mu_1)$ and $(X_2, \mathfrak{B}_2, \mu_2)$ be measure spaces and $T_1:X_1 \longrightarrow X_1$ , $T_2:X_2 \longrightarrow X_2$ be measure-preserving maps. Then, the product map $T_1 \times T_2 : X_1 \times X_2 \longrightarrow X_1 \times X_2$ , defined by
  
  is a measure-preserving transformation of $(T_1 \times T_2, \mathfrak{B}_1 \times \mathfrak{B}_1, \mu_1 \times \mu_2)$ .

Examples

• The identity map of a measure space $(X, \mathfrak{B}, \mu)$ is always measure-preserving.

• Let $G$ be a locally compact group. For each $a \in G$ , the transformation $T(g):=ag$ is measure-preserving relatively to any left Haar measure. Similarly, any right translation on $G$ preserves any right Haar measure.

• Every continuous surjective homomorphism between compact Hausdorff groups is measure-preserving relatively to the normalized Haar measure (see this entry).