# Atom (measure theory)

In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless.

## Definition

Given a measurable space ${\displaystyle (X,\Sigma )}$ and a measure ${\displaystyle \mu }$ on that space, a set ${\displaystyle A\subset X}$ in ${\displaystyle \Sigma }$ is called an atom if

${\displaystyle \mu (A)>0}$

and for any measurable subset ${\displaystyle B\subset A}$ with

${\displaystyle \mu (B)<\mu (A)}$

the set ${\displaystyle B}$ has measure zero.

If ${\displaystyle A}$ is an atom, all the subsets in the ${\displaystyle \mu }$-equivalence class ${\displaystyle [A]}$ of ${\displaystyle A}$ are atoms, and ${\displaystyle [A]}$ is called an atomic class. If ${\displaystyle \mu }$ is a ${\displaystyle \sigma }$-finite measure, there are countably many atomic classes.

## Examples

• Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra ${\displaystyle \Sigma }$ be the power set of X. Define the measure ${\displaystyle \mu }$ of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i=1,2, ..., 9, 10 is an atom.
• Consider the Lebesgue measure on the real line. This measure has no atoms.

## Atomic measures

A ${\displaystyle \sigma }$-finite measure ${\displaystyle \mu }$ on a measurable space ${\displaystyle (X,\Sigma )}$ is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a countable partition of ${\displaystyle X}$ formed by atoms.

## Discrete measures

An atomic measure ${\displaystyle \mu }$ is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent to say that ${\displaystyle \mu }$ is the weighted sum of countably many Dirac measures, that is, there is a sequence ${\displaystyle x_{1},x_{2},...}$ of points in ${\displaystyle X}$, and a sequence ${\displaystyle c_{1},c_{2},...}$ of positive real numbers (the weights) such that ${\textstyle \mu =\sum _{k=1}^{\infty }c_{k}\delta _{x_{k}}}$, which means that ${\textstyle \mu (A)=\sum _{k=1}^{\infty }c_{k}\delta _{x_{k}}(A)}$ for every ${\displaystyle A\in \Sigma }$. We can chose each point ${\displaystyle x_{k}}$ to be a common point of the atoms in the ${\displaystyle k}$-th atomic class.

A discrete measure is atomic but the inverse implication fails: take ${\displaystyle X=[0,1]}$, ${\displaystyle \Sigma }$ the ${\displaystyle \sigma }$-algebra of countable and co-countable subsets, ${\displaystyle \mu =0}$ in countable subsets and ${\displaystyle \mu =1}$ in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure ${\displaystyle \mu }$ is atomic but the intersection of the atoms in the unique atomic class is empty and ${\displaystyle \mu }$ can't be put as a sum of Dirac measures.

If every atom is equivalent to a singleton, ${\displaystyle \mu }$ is discrete iff it is atomic. In this case the ${\displaystyle x_{k}}$ above are the atomic singletons, so they are unique. Any finite measure in a separable metric space provided with the Borel sets satisfies this condition.[1]

## Non-atomic measures

A measure which has no atoms is called non-atomic measure or a diffuse measure. In other words, a measure ${\displaystyle \mu }$ is non-atomic if for any measurable set ${\displaystyle A}$ with ${\displaystyle \mu (A)>0}$ there exists a measurable subset ${\displaystyle B}$ of ${\displaystyle A}$ such that

${\displaystyle \mu (A)>\mu (B)>0.}$

A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set ${\displaystyle A}$ with ${\displaystyle \mu (A)>0}$ one can construct a decreasing sequence of measurable sets

${\displaystyle A=A_{1}\supset A_{2}\supset A_{3}\supset \cdots }$
such that
${\displaystyle \mu (A)=\mu (A_{1})>\mu (A_{2})>\mu (A_{3})>\cdots >0.}$

This may not be true for measures having atoms; see the first example above.

It turns out that non-atomic measures actually have a continuum of values. It can be proved that if ${\displaystyle \mu }$ is a non-atomic measure and ${\displaystyle A}$ is a measurable set with ${\displaystyle \mu (A)>0,}$ then for any real number ${\displaystyle b}$ satisfying

${\displaystyle \mu (A)\geq b\geq 0}$
there exists a measurable subset ${\displaystyle B}$ of ${\displaystyle A}$ such that
${\displaystyle \mu (B)=b.}$

This theorem is due to Wacław Sierpiński.[2][3] It is reminiscent of the intermediate value theorem for continuous functions.

Sketch of proof of Sierpiński's theorem on non-atomic measures. A slightly stronger statement, which however makes the proof easier, is that if ${\displaystyle (X,\Sigma ,\mu )}$ is a non-atomic measure space and ${\displaystyle \mu (X)=c,}$ there exists a function ${\displaystyle S:[0,c]\to \Sigma }$ that is monotone with respect to inclusion, and a right-inverse to ${\displaystyle \mu :\Sigma \to [0,c].}$ That is, there exists a one-parameter family of measurable sets ${\displaystyle S(t)}$ such that for all ${\displaystyle 0\leq t\leq t'\leq c}$

${\displaystyle S(t)\subseteq S(t'),}$
${\displaystyle \mu \left(S(t)\right)=t.}$
The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to ${\displaystyle \mu }$ :
${\displaystyle \Gamma :=\{S:D\to \Sigma \;:\;D\subseteq [0,c],\,S\;\mathrm {monotone} ,{\text{ for all }}t\in D\;(\mu (S(t))=t)\},}$
ordered by inclusion of graphs, ${\displaystyle \mathrm {graph} (S)\subseteq \mathrm {graph} (S').}$ It's then standard to show that every chain in ${\displaystyle \Gamma }$ has an upper bound in ${\displaystyle \Gamma ,}$ and that any maximal element of ${\displaystyle \Gamma }$ has domain ${\displaystyle [0,c],}$ proving the claim.