In other words, if is invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) by then or .
Suppose is a probability space and is a measure-preserving transformation. If is an invariant (http://planetmath.org/InvariantByAMeasurePreservingTransformation) measurable subset, with , then is also invariant and . Thus, in this situation, we can study the transformation by studying the two simpler transformations and in the spaces and , respectively.
The transformation is ergodic precisely when cannot be decomposed into simpler transformations. Thus, ergodic transformations are the measure-preserving transformations, in the sense described above.
Remark: When the invariant has measure we can ignore it (as usual in measure theory), as its presence does not affect significantly. Thus, the study of is not simplified when restricting to .
|Date of creation||2013-03-22 12:19:38|
|Last modified on||2013-03-22 12:19:38|
|Last modified by||asteroid (17536)|