Non-atomic measure
From Encyclopedia of Mathematics
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=12299
Non-atomic measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-atomic_measure&oldid=12299
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