operator induced by a measure preserving map
theorem \PMlinkescapephraseinduced \PMlinkescapephraseproperty \PMlinkescapephraseproperties \PMlinkescapephraseside
1 Induced Operators
The operator is called the by .
Many ideas in ergodic theory can be explored by studying this operator.
2 Basic Properties
The following are clear:
3 Preserving Integrals
Theorem 1 - If then , where if one side does not exist or is infinite, then the other side has the same property.
4 Induced Isometries
It can further be seen that a measure-preserving transformation induces an isometry between -spaces (http://planetmath.org/LpSpace), for .
Theorem 2 - Let . We have that and, moreover,
Thus, when restricted to -spaces, is called the isometry induced by .
|Title||operator induced by a measure preserving map|
|Date of creation||2013-03-22 17:59:19|
|Last modified on||2013-03-22 17:59:19|
|Last modified by||asteroid (17536)|
|Defines||isometry induced by a measure preserving map|