Definition:Atom of Measure

This page is about atoms in the context of measures. For other uses, see Atom.

Definition

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

An element $x \in X$ is said to be an atom (of $\mu$) if and only if:

$(1): \quad \set x \in \Sigma$

$(2): \quad \map \mu {\set x} > 0$

This article is complete as far as it goes, but it could do with expansion. In particular: There is a different (but probably equivalent) definition in 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) and 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.), which could be documented. Also in 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Definition:Diffuse Measure: a measure without atoms

Linguistic Note

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.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 6$: Problem $5$