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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.
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