Entropy of a measurable decomposition

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$\xi$ 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


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


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


[a1] I.P. [I.P. Kornfel'd] Cornfel'd, S.V. Fomin, Ya.G. Sinai, "Ergodic theory" , Springer (1982) (Translated from Russian)
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Entropy of a measurable decomposition. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article