Entropy of a measurable decomposition

$\xi$ 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)