Atom (measure theory)

An atom of a measure $\mu$ is a measurable set $A$ of positive measure for which every measurable $B\subseteq A$ has either $\mu (B)=0$ or $\mu (B)=\mu (A)$. A singleton of positive mass is an example. A transport map cannot split the mass of a singleton among several destinations.

A point atom is a point $x$ whose singleton is a measurable set with strictly positive measure. That singleton is an atom of a measure. This is more specific than a general atom of a measure, which is defined as a positive-measure set that cannot be divided into two measurable sets of positive measure.

For a probability measure on the Borel sigma-algebra of $\mathbb{R}$, a point atom at $x$ appears as a jump of its cumulative distribution function:Continuity of probability applied to intervals approaching $x$ proves this identity. Keeping such jumps is essential when deciding whether an endpoint belongs to an integration region.

For a finite measure on a measurable space whose singletons are measurable sets, the set of point atoms of a measure is at most countable. For each positive integer $m$, let $D_m$ consist of the points with singleton measure at least $1/m$. If $D_m$ contained more than $m\mu (\Omega )$ distinct points, finite additivity on those disjoint singletons would exceed the total measure. Thus each $D_m$ is finite.

Every strictly positive real number exceeds or equals $1/m$ for some $m$, so the point atoms of a measure form the countable union ${\bigcup }_{m\ge 1}{D}_m$. This proves that the set is countable. It also shows that a cumulative distribution function has only countably many jumps. The result does not say the measure is concentrated on those points: a non-atomic measure may assign them total measure zero.

A non-atomic measure contains no atom of a measure. For a finite Borel measure on a Polish space, this is equivalent to giving every singleton mass zero. For a probability measure on $\mathbb{R}$, it is also equivalent to continuity of its cumulative distribution function.

The range of a finite-dimensional, finite, countably additive, atomless vector measure is compact and convex. In particular, an atomless measure restricted to a measurable set $E$ can realize every mass between zero and $\mu (E)$. This allows exact balancing by small disjoint consumer cohorts in an exchange economy.