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.