where is, as usual, the set of points such that .
Measure-preserving transformations between the same measure space are sometimes called of the measure space.
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 and be measure spaces and and their completions. If is measure-preserving, then so is .
Let and be measure spaces and , be measure-preserving maps. Then, the product map , defined by
is a measure-preserving transformation of .
The identity map of a measure space is always measure-preserving.
|Date of creation||2013-03-22 12:19:41|
|Last modified on||2013-03-22 12:19:41|
|Last modified by||asteroid (17536)|
|Defines||invertible measure-preserving transformation|
|Defines||endomorphism of a measure space|