# Entropy of a measurable decomposition

From Encyclopedia of Mathematics

*$\xi$ of a space with a normalized measure $(X,\mu)$*

A concept defined as follows. If the elements of $\xi$ having measure zero form in total a set of positive measure, then the entropy of $\xi$ is $H(\xi)=\infty$; otherwise

$$H(\xi)=-\sum\mu(C)\log\mu(C),$$

where the sum is taken over all elements of $\xi$ of positive measure. The logarithm is usually to the base 2.

#### Comments

Instead of "measurable decomposition" the phrase "measurable partitionmeasurable partition" is often used, cf. [a1].

#### References

[a1] | I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian) |

**How to Cite This Entry:**

Entropy of a measurable decomposition.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Entropy_of_a_measurable_decomposition&oldid=33192

This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article