You are here
Home ›metric entropy
Primary tabs
metric entropy
Let be a probability space, and a measure-preserving transformation. The entropy of with respect to a finite measurable partition is
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.
Remarks.
1. 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.
Mathematics Subject Classification
28D20 Entropy and other invariants37A35 Entropy and other invariants, isomorphism, classification
- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)
- Other useful stuff
Recent Activity
new question: pure subgroups by lvoyster
new correction: Typo in M\"obius function? by Aleph Zero
new collection: analytic number theory by Aleph Zero
May 20
new question: Taylor's Series Query! by unlord
new question: Laplace transform by J
new question: Residue Calculus by J
May 19
new Education: Project: PlanetMath Outlines Series by unlord
May 17
new image: sinx_approx.png by jeremyboden
new image: approximation_to_sinx by jeremyboden
new image: approximation_to_sinx by jeremyboden


