Definition of Haar measures

Let be a locally compact topological group, and denote by $\mathcal{B}$ the sigma algebra generated by the closed compact subsets of . A left Haar measure on is a measure $\mu$ on $\mathcal{B}$ which is:

outer regular on all sets $B \in \mathcal{B}$
inner regular on all open sets $U \in \mathcal{B}$
finite on all compact sets $K \in \mathcal{B}$
invariant under left translation: $\mu(gB) = \mu(B)$ for all sets $B \in \mathcal{B}$
nontrivial: $\mu(B) > 0$ for all non-empty open sets $B \in \mathcal{B}$ .

A right Haar measure on is defined similarly, except with left translation invariance replaced by right translation invariance ( $\mu(Bg) = \mu(B)$ for all sets $B \in \mathcal{B}$ ). A bi-invariant Haar measure is a Haar measure that is both left invariant and right invariant.

Existence of Haar measures

For any discrete topological group , the counting measure on is a bi-invariant Haar measure. More generally, every locally compact topological group has a left Haar measure $\mu$ , which is unique up to scalar multiples. In addition, also admits a right Haar measure, and for an abelian group the left and right Haar measures are always equal. The Haar measure plays an important role in the development of Fourier analysis and representation theory on locally compact groups such as Lie groups and profinite groups.

"Haar measure" is owned by djao.

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Also defines:

left Haar measure, right Haar measure, bi-invariant Haar measure

Attachments:: modular function (Definition) by asteroid

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Cross-references: profinite groups, Lie groups, groups, theory, representation, development, abelian group, addition, scalar multiples, counting measure, discrete, right, translation, invariant, compact sets, finite, open sets, inner regular, outer regular, measure, compact subsets, closed, generated by, sigma algebra, topological group, locally compact
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This is version 6 of Haar measure, born on 2002-05-28, modified 2008-05-08.
Object id is 2959, canonical name is HaarMeasure.
Accessed 10447 times total.

Classification:

AMS MSC:

28C10 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological groups, Haar measures, invariant measures)

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