Non-atomic measure

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.