Haar measure, the Peter-Weyl theorem and compact abelian groups

Haar measure

We restrict our attention to groups that are locally compact Hausdorff and \sigma-compact.

Definition. (Radon measure) Let X be a \sigma-compact locally compact Hausdorff space. The Borel \sigma-algebra \mathcal{B}[X] on X is the \sigma-algebra generated by the open subsets of X. A Borel measure is a countably additive nonnegative measure \mu:\mathcal{B}[X]\to[0,+\infty] on the Borel \sigma-algebra. A Radon measure is a Borel measure  obeying three additional axioms:

(i) (Local finiteness) One has \mu(K)<\infty for every compact set K.

(ii) (Inner regularity) One has \mu(E)=\sup_{K\subset E,K\ \text{compact}}\mu(K) for every Borel measureable set E.

(iii) (Outer regularity)  One has \mu(E)=\inf_{U\supset E,U\ \text{open}}\mu(U) for every Borel measureable set E.

Definition. (Haar measure) Let G=(G,\cdot) be a \sigma-compact locally Hausdorff group. A Radon measure \mu is left-invariant (resp. right-invariant) if one has \mu(gE)=\mu(E) (resp. \mu(Eg)=\mu(E) ) for all latex g\in G$ and Borel measureable sets E. A left-invariant Haar measure is a nonzero Radon measure which is left-invariant.

Theorem. (Riesz representation theorem). Let X be a \sigma-compact locally compact Hausdorff space. Then to every liner functional I:C_c(X)\to \mathbf{R} whihc is nonnegative (thus I(f)\geq 0 whenever f\geq 0), one can associate a unique Radon measure \mu such that I(f)=\int_X f\,d\mu for all f\in C_c(X). Conversely, for each Radon measure \mu, the functional I_\mu:f\mapsto \int_X f\,d\mu is a nonnegative linera functional on C_c(X).