Haar measure


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Haar measure

[′här ‚mezh·ər]
(mathematics)
A measure on the Borel subsets of a locally compact topological group whose value on a Borel subset U is unchanged if every member of U is multiplied by a fixed element of the group.
References in periodicals archive ?
Subgroups have nice properties and a closed subgroup of a locally compact abelian group is itself a locally compact abelian group, with Haar measure [m.
When H is a topologically closed subset of G (a technicality that is satisfied in all that follows), G/H is also a locally compact abelian group, with Haar measure [m.
First, it has an essentially unique Haar measure and, secondly, it is associated with a dual group.
Haar measure is a very general extension of a 'length' or 'area', which it turns out is in many respects like Lebesgue measure (a mathematical name for ordinary length, area or volume in Euclidean space).
The uniqueness of Haar measure implies that the integral over G of a function f (in [L.
The dual group [GAMMA] is also locally compact and abelian, with its Haar measure denoted by [m.
implying that f^ [disjunction]] (x) = f(x) except for a set of Haar measure 0 and often written for convenience more simply as f^ [disjunction]] ~ f.
The reciprocity relation (15) ensures that if the Haar measure for a compact group is chosen to be unity, the Haar measure on the discrete dual group has to be counting measure (cf.
Coset decomposition will now be used to express the Haar measure of the transversal [OMEGA] in terms of the Haar measures of the compact quotient [GAMMA]/[LAMBDA] and the point {0} in [LAMBDA].
Let A be a subset of [GAMMA] with positive and finite Haar measure m[GAMMA](A) (the set A corresponds to the set of frequencies of the signals being considered).
An almost disjoint translates set is almost contained in a transversal, [OMEGA] say, of [GAMMA]/[LAMBDA], so that A [union] ([GAMMA] \ [OMEGA]) is of Haar measure zero and m[GAMMA](A) [less than or equal to] m[GAMMA]([OMEGA]).
He covers the basic concepts, Gaussian measures, dynamical system, Borel product-measures, invariant Borel measures, quasi-invariant Radon measures, partial analogies of Lebegues measures, essential uniqueness, the Erdos-Sierpinski duality principle, strict transivity properties, invariant extensions of Haar measures, separated families of probability measures, an Ostrogradsky formula, and generalized Fourier series.