Editing Locally compact space (section)

=== Locally compact groups ===
The notion of local compactness is important in the study of [[topological group]]s mainly because every Hausdorff [[locally compact group]] ''G'' carries natural [[Measure theory|measures]] called the [[Haar measure]]s which allow one to [[integral|integrate]] [[measurable function]]s defined on ''G''.
The [[Lebesgue measure]] on the [[real line]] <math>\R</math> is a special case of this.

The [[Pontryagin dual]] of a [[topological abelian group]] ''A'' is locally compact [[if and only if]] ''A'' is locally compact.
More precisely, Pontryagin duality defines a self-[[Duality (category theory)|duality]] of the [[category theory|category]] of locally compact abelian groups.
The study of locally compact abelian groups is the foundation of [[harmonic analysis]], a field that has since spread to non-abelian locally compact groups.