Corollary 7. similar way a left invariant Haar measure is de ned.

Let Gbe a topological group, let be a Haar measure on G, and de ne 0(A) = 0 A 1.

L and ! a left Haar measure need not be a right Haar measure. R in previous example satis es !

DepartmentofMathematical Sciences,UniversityofCopenhagen MarcusD.DeChiffre Supervisor: MagdalenaMusat June62011 left) Haar measure on G. We now look at some applications of Haar measure to the study of representations of compact groups.

(5)The Haar measure is an integral tool in representation theory.

L = ! Proposition 4.1. Example 2.1. (6) The proof of existence of Haar measure for compact Lie groups is much cleaner: in that setting, we value form over function.

Example.

Anatole Katok, Jean-Paul Thouvenot, in Handbook of Dynamical Systems, 2006. L = y 2dxdy is the left Haar measure on G, and! Institutformatematiskefag, KøbenhavnsUniversitet BachelorThesisinMathematics. In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.. (Uniqueness of Haar measure). If J is the inversion in G, a measure is a right invariant Haar measure if and only if J is left-invariant Haar measure. Then, is a left (resp. The Haar measure Bachelorprojektimatematik. R = y 1dxdy is the right Haar measure on G. One can check that ! It follows that the Haar measure of each compact set is nite and the Haar measure of each open set is strictly positive. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Theorem 8. right) Haar measure i 0is a right (resp. Consider an automorphism A of a compact Abelian group G.It preserves Haar measure χ and the Koopman operator maps characters into characters. obtain left Haar measure from right Haar measure, and vice versa. …

Consider G= ˆ y x 0 1 jx;y2R;y>0 ˙; then one can check that up to a multiplicative constant,!

We use Γ to denote a compact group, µ its Haar measure, V a finite dimensional complex vector space with inner product and Hom(V ) the set of linear maps on V .