By Klaus Bichteler

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**Additional resources for Integration Theory, with Special Attention to Vector Measures**

**Example text**

5 Let G be a compact and Hausdorff group. Then every irreducible and unitary representation of G on a separable and complex Hilbert space is finitedimensional. Proof. Let 1r : G - t U(X) be an irreducible and unitary representation of G on X. Let cp E X be such that Ilcpli = 1. L(g) , x, Y E X. 2) Then Tcp : X - t X is a bounded linear operator. L(G)llxllllyll· 5. Unitary Representations Moreover, Tcp : X --+ 37 X is a positive operator. Indeed, let x E X. } be an orthonormal basis for X. Then, by Fubini's theorem, Parseval's identity, II

K and h f/. K, then it is obvious that f(g) = f(h) = O. Hence we have proved that f is left uniformly continuous on G. The proof for right uniform continuity is similar and hence omitted. 0 A complex-valued function on G that is left and right uniformly continuous on G is said to be uniformly continuous on G. 4 Haar Measures and Modular Functions We assume a basic knowledge of measure theory on locally compact and Hausdorff topological spaces, which can be found in [73] by Royden and [74] by Rudin, among others.

5). 11. :i. be the modular function on a locally compact and Hausdorff group G. 5). Proof. L(h) = 1. 23) 32 4. L(B- 1) = jj(B). 11. Suppose that G is unimodular. (g) = 1 for all gin G. Let B be a Borel subset of G. L(B). L. L. (go) =I- 1 for some go in G. (g-1) =I- 1 for all 9 in U. (g-1) > 1 for all 9 in U. 12, and this is a contradiction. , G is unimodular. 13 Every abelian, locally compact and Hausdorff group is unimodu- lar. Proof. Let 9 E G and let B be any Borel subset of G. L(B). 24) 33 4.