Let Ω=(X,**B**,μ) and Ψ=(Y,**C**,ν) be measure spaces. A transformation (i.e. function)

Φ : X → Y

is called

*measure-preserving* if its

preimage preserves measures. That is, if for every T⊆Y for which T∈

**C**:

- Φ
^{-1}(T)∈**B** (the preimage of a measurable set is measurable)
- μ(Φ
^{-1}(T)) = ν(T) (the preimage has the same measure)

The use of preimage, rather than image, in the definition appears odd. But it is very necessary. For instance, let Ω be the uniform probability space on the square [0,1]×[0,1], and let Τ be the uniform probability space on the segment [0,1]. Then the projection π:[0,1]×[0,1]→[0,1], defined by π(a,b)=b, preserves measure.

On the other hand, consider the set A=[a,a']×[0,1] (a'≥a), which has measure a'-a. We always have that π(A)=[0,1], which has measure 1. By examining the measure of the preimage of π, rather than its image, we neatly sidestep the issue and get a workable definition for a measure-preserving transformation, that works even for functions that are not "almost always" one-to-one.

A similar phenomenon appears for many definitions: immediate examples are continuous functions and all types of homomorphisms. In fact, a measure-preserving transformation *is* a homomorphism of measure spaces.