Difference between revisions of "Entropy of a measurable decomposition"
From Encyclopedia of Mathematics
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| + | ''$\xi$ of a space with a normalized measure $(X,\mu)$'' | ||
| − | A concept defined as follows. If the elements of | + | 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 | + | where the sum is taken over all elements of $\xi$ of positive measure. The logarithm is usually to the base 2. |
Latest revision as of 08:36, 29 August 2014
$\xi$ 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=12139
Entropy of a measurable decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Entropy_of_a_measurable_decomposition&oldid=12139
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article