# 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 $C_c(X)$.