invariant by a measure-preserving transformation
invariant \PMlinkescapephraseproperties \PMlinkescapephraseproperty
Let be a set and a transformation of .
The notion of invariance by we are about to describe is stronger than the usual notion of invariance (http://planetmath.org/invariant), and is especially useful in ergodic theory. Thus, in most applications, is a measure space and is a measure-preserving transformation. Nevertheless, the definition of invariance and its properties are general and do not require any such assumptions.
Definition - A subset is said to be invariant by , or -invariant, if .
The fundamental property of this concept is the following: if is invariant by , then so is .
Thus, when is invariant by we obtain by restriction two well-defined transformations
Hence, the existence of an allows one to decompose the set into two disjoint subsets and study the transformation in each of these subsets.
Remark - When is a measure-preserving transformation in a measure space one usually restricts the notion of invariance to measurable subsets .
|Title||invariant by a measure-preserving transformation|
|Date of creation||2013-03-22 18:04:15|
|Last modified on||2013-03-22 18:04:15|
|Last modified by||asteroid (17536)|