Namespaces
Variants
Actions

Difference between revisions of "Non-atomic measure"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(→‎References: Feller: internal link)
Line 11: Line 11:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley  (1971)  pp. 135</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''2''' , Wiley  (1971)  pp. 135</TD></TR></table>

Revision as of 18:32, 26 April 2012

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