 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.

In measure theory, an **atom** is a specific type of measurable set associated with a measure. An atom is a measurable set that contains "mass" in the sense that it cannot be subdivided into smaller measurable sets with lower measure.