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.

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

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.

Atomic measures

A $\sigma$ -finite measure $\mu$ on a measurable space $(X,\Sigma )$ 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 $X$ formed by atoms.

Discrete measures

An atomic measure $\mu$ is called discrete if the intersection of the atoms of any atomic class is non empty. It is equivalent to say that $\mu$ is the weighted sum of countably many Dirac measures, that is, there is a sequence $x_{1},x_{2},...$ of points in $X$ , and a sequence $c_{1},c_{2},...$ 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 $A\in \Sigma$ . We can chose each point $x_{k}$ to be a common point of the atoms in the $k$ -th atomic class.

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

If every atom is equivalent to a singleton, $\mu$ is discrete iff it is atomic. In this case the $x_{k}$ 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.

Non-atomic measures

A measure which has no atoms is called non-atomic measure or a diffuse measure. In other words, a measure $\mu$ 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 $\mu$ 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 $0\leq t\leq t'\leq c$ $S(t)\subseteq S(t'),$ $\mu \left(S(t)\right)=t.$ The proof easily follows from Zorn's lemma applied to the set of all monotone partial sections to $\mu$ :
$\Gamma :=\{S:D\to \Sigma \;:\;D\subseteq [0,c],\,S\;\mathrm {monotone} ,{\text{ for all }}t\in D\;(\mu (S(t))=t)\},$ ordered by inclusion of graphs, $\mathrm {graph} (S)\subseteq \mathrm {graph} (S').$ 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.