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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Thus, on ([OMEGA], b([OMEGA])), the probability Haar measure [m.sub.H] can be defined, and this leads to the probability space ([OMEGA], b([OMEGA]), [m.sub.H]).
Then [Q.sub.N,w] converges weakly to the Haar measure [m.sub.h] as N [right arrow] [infinity].
Since the right-hand side of (2.3) is the Fourier transform of the Haar measure [m.sub.H], by a continuity theorem for probability measures on compact groups, we obtain that [Q.sub.N,w] converges weakly to [m.sub.H] as N [right arrow] [infinity].
where dh is the Haar measure of H with vol([K.sub.H]; dh) = 1.
Here dt is the Haar measure of [T.sub.H] with vol([T.sub.H] [intersection] [K.sub.H]; dt) = 1.
They cover Lebesgue measure in Euclidean space, measures on metric spaces, topological groups, Banach and measure, compact groups have a Haar measure, applications, Haar measure on locally compact groups, metric invariance and Haar measure, Steinlage on Haar measure, and Oxtoby's view of Haar measure.
Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the left Haar measure of U.
which shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the right Haar measure of U.
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.sub.H].
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.sub.G/H], and plays an important role in the abstract theory.
Then, on (G, b(G)), the probability Haar measure [mu] can be defined.
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.

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