Definition:Atom of Sigma-Algebra

This page is about Atom in the context of Sigma-Algebra. For other uses, see Atom.

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Let $E \in \Sigma$ be non-empty.

$E$ is said to be an atom (of $\Sigma$) if and only if it satisfies:

$\forall F \in \Sigma: F \subsetneq E \implies F = \O$

Thus, atoms are the minimal non-empty sets in $\Sigma$ with respect to the subset ordering.

The word atom comes from the Greek ἄτομον, meaning unbreakable or indecomposable.

It is pronounced with a short a, as at-tom, as opposed to ay-tom.