nonatomic

Nonatomic, in measure theory and probability, describes a measure space with no atoms. An atom is a measurable set A with μ(A) > 0 such that every measurable subset B ⊆ A has μ(B) ∈ {0, μ(A)}. A measure μ is nonatomic if it has no atoms; equivalently, for every measurable set A with μ(A) > 0 there exists a measurable B ⊆ A with 0 < μ(B) < μ(A). A probability space is nonatomic when its probability measure is nonatomic.

Examples include the Lebesgue measure on the unit interval [0,1], which assigns positive measure to intervals and allows division into smaller positive-measure subsets. By contrast, the counting measure on any infinite set is not nonatomic, since every singleton has positive measure and cannot be subdivided within a finite-measure set.

Properties and implications: In a nonatomic space, for any event of positive probability you can find a sub-event with arbitrarily smaller positive probability, enabling construction of random variables with continuous distributions and uniform distributions on subintervals. Many standard probabilistic and analytical results assume nonatomicity; such spaces are often described as diffuse measures.

See also: atom, atomless, diffuse measure, probability space, Lebesgue measure.