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Non-atomic measure

From Encyclopedia of Mathematics
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A measure on a measurable space for which there are no atoms of positive measure, i.e. sets with for which and imply .


Comments

An atom in a measure space is a set for which i) ; and ii) and imply either or . See also Atom.

A measure space is called non-atomic if no element of is an atom. In probability theory measure spaces build up completely from atoms, i.e. using atomic measures, frequently occur, cf. Atomic distribution.

A probability decomposes as a sum , , where is an atomic distribution and a continuous distribution, i.e. a non-atomic one. This goes by the name Jordan decomposition theorem.

References

[a1] W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971) pp. 135
How to Cite This Entry:
Non-atomic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-atomic_measure&oldid=25537
This article was adapted from an original article by N.N. Vorob'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article