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{{MSC|03E04}} ''in set theory'' 

{{MSC|28A}} ''in measure theory''

[[Category:Classical measure theory]]

[[Category:Set theory]]

{{TEX|done}}

==Ordered sets==
A  minimal non-zero element of a [[partially ordered  set]] with a zero $0$, i.e. a [[covering element]] of $0$; an element $p > 0$ such that $0<x\leq p$  implies $x=p$.

==Measure algebras==
For the definition and relevance in the theory of measure algebras we refer to [[Measure algebra (measure theory)|Measure algebra]].

==Classical measure theory==
===Definition===
Let  $\mu$ be a (nonnegative) [[Measure|measure]] on a [[Algebra of  sets|$\sigma$-algebra]] $\mathcal{S}$ of subsets of a set $X$. An  element $a\in \mathcal{S}$ is called an ''atom'' of $\mu$ if
*$\mu (A)>0$;
*For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$
(cp. with Section IV.9.8 of {{Cite|DS}} or {{Cite|Fe}}).

'''Remark''' 
If  we denote by $\mathcal{N}$ the null sets and consider the standard  quotient measure algebra $(\mathcal{S}/\mathcal{N}, \mu)$, then any atom  of such quotient measure algebra corresponds to an equivalence class of  atoms of $\mu$.
===Atomic measures===
A σ-finite measure $\mu$ is called ''atomic'' if there is a partition of $X$ into  countably many elements of $\mathcal{A}$ which are either atoms or null  sets. An atomic probability measure is often called ''atomic  distribution''. Examples of atomic distributions are the [[Discrete  distribution|discrete distributions]].

===Nonatomic measures===
A measure $\mu$ is called ''nonatomic'' if it has no atoms.

===Jordan decomposition===
If  $\mu$ is $\sigma$-finite, it is possible to decompose $\mu$ as  $\mu_a+\mu_{na}$, where $\mu_a$ is an atomic measure and $\mu_{na}$ is a  nonatomic measure. In case $\mu$ is a probability measure, this means  that $\mu$ can be written as $p \mu_a + (1-p) \mu_{na}$, where $p\in  [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ a  nonatomic probability measure (see {{Cite|Fe}}), which is sometimes  called a [[Continuous distribution|continuous distribution]]. This  decomposition is sometimes called ''Jordan decomposition'', although  several authors use this name in other contexts, see [[Jordan  decomposition]].
===Measures in the euclidean space===
If  $\mu$ is a $\sigma$-finite measure on the [[Borel set|Borel  $\sigma$-algebra]] of $\mathbb R^n$, then it is easy to show that, for  any atom $B$ of $\mu$ there is a point $x\in B$ with the property that  $\mu (B) = \mu (\{x\})$. Thus such a measure is atomic if and only if it  is the countable sum of [[Delta-function|Dirac deltas]], i.e. if there  is an (at most) countable set $\{x_i\}\subset \mathbb R^n$ and an (at  most) countable set $\{\alpha_i\}\subset ]0, \infty[$ with the property  that
\[
\mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set } A\, .
\]

===Sierpinski's theorem===
A  nonatomic measure takes a continuum of values. This is a corollary of  the following Theorem due to Sierpinski (see {{Cite|Si}}):

'''Theorem'''
If  $\mu$ is a nonatomic measure on a $\sigma$-algebra $\mathcal{A}$ and  $A\in \mathcal{A}$ an element such that $\mu (A)>0$, then for every  $b\in [0, \mu (A)]$ there is an element $B\in \mathcal{A}$ with  $B\subset A$ and $\mu (B) = b$.

==Comment==
By  a natural extension of meaning, the term atom is also used for an  object of a category having no subobjects other than itself and the null  subobject (cf. [[Null object of a category|Null object of a  category]]).

==References==
{|
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|valign="top"|{{Ref|DS}}||       N. Dunford, J.T. Schwartz,  "Linear operators. General theory",      '''1''', Interscience (1958).  {{MR|0117523}} {{ZBL|0635.47001}}
|-
|valign="top"|{{Ref|Fe}}||  W. Feller, "An introduction to  probability theory and its   applications"|"An introduction to  probability theory and its   applications", '''2''', Wiley (1971).
|-
|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory", Princeton Univ. Press (1963). {{MR|0203748}} {{ZBL|0108.14202}}
|-
|valign="top"|{{Ref|Si}}||  W. Sierpinski, "Sur le fonctions d'enseble additives et continuoes",  '''3''', Fund. Math. (1922) pp. 240-246.
|-
|}