# 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 $(X,\Sigma )$  and a measure $\mu$  on that space, a set $A\subset X$  in $\Sigma$  is called an atom if

$\mu (A)>0$

and for any measurable subset $B\subset A$  with

$\mu (B)<\mu (A)$

the set $B$  has measure zero.

## Examples

• Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra $\Sigma$  be the power set of X. Define the measure $\mu$  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.

## Non-atomic measures

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

$\mu (A)>\mu (B)>0.\,$

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

$A=A_{1}\supset A_{2}\supset A_{3}\supset \cdots$

such that

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

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 μ is a non-atomic measure and A is a measurable set with $\mu (A)>0,$  then for any real number b satisfying

$\mu (A)\geq b\geq 0\,$

there exists a measurable subset B of A such that

$\mu (B)=b.\,$

This theorem is due to Wacław Sierpiński. 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 $(X,\Sigma ,\mu )$  is a non-atomic measure space and $\mu (X)=c$ , there exists a function $S:[0,c]\to \Sigma$  that is monotone with respect to inclusion, and a right-inverse to $\mu :\Sigma \to [0,\,c]$ . That is, there exists a one-parameter family of measurable sets S(t) such that for all $0\leq t\leq t'\leq c$

$S(t)\subset S(t'),$
$\mu \left(S(t)\right)=t.$

The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to $\mu$  :

$\Gamma :=\{S:D\to \Sigma \;:\;D\subset [0,\,c],\,S\;\mathrm {monotone} ,\forall t\in D\;(\mu \left(S(t)\right)=t)\},$

ordered by inclusion of graphs, $\mathrm {graph} (S)\subset \mathrm {graph} (S').$  It's then standard to show that every chain in $\Gamma$  has an upper bound in $\Gamma$ , and that any maximal element of $\Gamma$  has domain $[0,c],$  proving the claim.