Atom

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

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

## Contents

* 1 Ordered sets
* 2 Measure algebras
* 3 Classical measure theory
* 3.1 Definition
* 3.2 Atomic measures
* 3.3 Nonatomic measures
* 3.4 Jordan decomposition
* 3.5 Measures in the euclidean space
* 3.6 Sierpinski's theorem
* 4 Set theory
* 5 Comment
* 6 References

## Ordered sets[edit]

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[edit]

For the definition and relevance in the theory of measure algebras we refer to Measure algebra.

## Classical measure theory[edit]

### Definition[edit]

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[edit]

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[edit]

A measure $\mu$ is called nonatomic if it has no atoms.

### Jordan decomposition[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

[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