Knowlpedia

Definition

Atom of a measure

A positive-measure measurable set with no measurable part of intermediate measure.

For a (X,Σ,μ)(X,\Sigma,\mu), an atom is a measurable set AA of positive measure such that every measurable B⊆AB\subseteq A satisfies μ(B)=0\mu(B)=0 or μ(B)=μ(A)\mu(B)=\mu(A).

Examples

A point of positive probability is an atom. On a finite uniform space, every singleton is an atom. Lebesgue measure on an interval has no atoms: each positive-measure set can be divided into two measurable parts of positive measure.