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 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.