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A minimal non-zero element of a [[Partially ordered set|partially ordered set]] with a zero 0, i.e. an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013850/a0138501.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013850/a0138502.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013850/a0138503.png" />.
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{{MSC|03E04}} ''in set theory''
  
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{{MSC|28A}} ''in measure theory''
  
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[[Category:Classical measure theory]]
  
====Comments====
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[[Category:Set theory]]
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]]).
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{{TEX|done}}
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==Ordered sets==
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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 $x=p$.
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==Measure algebras==
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For the definition and relevance in the theory of measure algebras we refer to [[Measure algebra (measure theory)|Measure algebra]].
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==Classical measure theory==
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===Definition===
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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
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*$\mu (A)>0$;
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*For every $B\in \mathcal{S}$ with $B\subset A$ either $\mu (B)=0$ or $\mu (B)=\mu (A)$
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(cp. with Section IV.9.8 of {{Cite|DS}} or {{Cite|Fe}}).
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'''Remark'''
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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 $\mu$.
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===Atomic measures===
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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]].
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===Nonatomic measures===
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A measure $\mu$ is called ''nonatomic'' if it has no atoms.
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===Jordan decomposition===
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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]].
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===Measures in the euclidean space===
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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
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\[
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\mu (A) = \sum_{x_i\in A} \alpha_i \qquad \mbox{for every Borel set } A\, .
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\]
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===Sierpinski's theorem===
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A  nonatomic measure takes a continuum of values. This is a corollary of  the following Theorem due to Sierpinski (see {{Cite|Si}}):
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'''Theorem'''
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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 $\mu (B) = b$.
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==Set theory==
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In some models of set theory, an atom or ''urelement'' is an entity which may be an element of a set, but which itself can have no elements.  Zermelo–Fraenkel [[axiomatic set theory]] with atoms is denoted ZFA (see {{Cite|Je}}).
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==Comment==
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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]]).
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==References==
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{|
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|-
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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}}
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|-
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|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).
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|-
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|valign="top"|{{Ref|Je}}||  T. Jech, "Set theory. The third millennium edition, revised and expanded" Springer Monographs in Mathematics (2003). {{ISBN|3-540-44085-2}} {{ZBL|1007.03002}}
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|-
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|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory", Princeton Univ. Press (1963). {{MR|0203748}} {{ZBL|0108.14202}}
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|-
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|valign="top"|{{Ref|Si}}||  W. Sierpiński, "Sur les fonctions d’ensemble additives et continues",  '''3''', Fund. Math. (1922) pp. 240-246 {{ZBL|48.0279.04}}
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|-
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|}

Latest revision as of 19:18, 14 November 2023

2020 Mathematics Subject Classification: Primary: 03E04 [MSN][ZBL] in set theory

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL] in measure theory

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.

Classical measure theory

Definition

Let $\mu$ be a (nonnegative) measure on a $\sigma$-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 [DS] or [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 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 [Fe]), which is sometimes called a 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 $\sigma$-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 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 [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 $\mu (B) = b$.

Set theory

In some models of set theory, an atom or urelement is an entity which may be an element of a set, but which itself can have no elements. Zermelo–Fraenkel axiomatic set theory with atoms is denoted ZFA (see [Je]).

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).

References

[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958). MR0117523 Zbl 0635.47001
[Fe] "An introduction to probability theory and its applications", 2, Wiley (1971).
[Je] T. Jech, "Set theory. The third millennium edition, revised and expanded" Springer Monographs in Mathematics (2003). ISBN 3-540-44085-2 Zbl 1007.03002
[Lo] M. Loève, "Probability theory", Princeton Univ. Press (1963). MR0203748 Zbl 0108.14202
[Si] W. Sierpiński, "Sur les fonctions d’ensemble additives et continues", 3, Fund. Math. (1922) pp. 240-246 Zbl 48.0279.04
How to Cite This Entry:
Atom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Atom&oldid=14251
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article