where is the entropy of a partition and denotes the join of partitions. The above limit always exists, although it can be . The entropy of is then defined as
with the supremum taken over all finite measurable partitions. Sometimes is called the metric or measure theoretic entropy of , to differentiate it from topological entropy.
There is a natural correspondence between finite measurable partitions and finite sub--algebras of . Each finite sub--algebra is generated by a unique partition, and clearly each finite partition generates a finite -algebra. Because of this, sometimes is called the entropy of with respect to the -algebra generated by , and denoted by . This simplifies the notation in some instances.
|Date of creation||2013-03-22 14:31:59|
|Last modified on||2013-03-22 14:31:59|
|Last modified by||Koro (127)|
|Synonym||measure theoretic entropy|