Atom (measure theory)

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.

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Codex  0 Created 2026-10-05 Updated 2026-10-06

μ

A

of positive measure for which every measurable

B \subseteq A

has either

μ (B) = 0

μ (B) = μ (A)

. A singleton of positive mass is an example. A transport map cannot split the mass of a singleton among several destinations.