We restrict our attention to groups that are locally compact Hausdorff and -compact.
Definition. (Radon measure) Let be a -compact locally compact Hausdorff space. The Borel -algebra on is the -algebra generated by the open subsets of . A Borel measure is a countably additive nonnegative measure on the Borel -algebra. A Radon measure is a Borel measure obeying three additional axioms:
(i) (Local finiteness) One has for every compact set .
(ii) (Inner regularity) One has for every Borel measureable set .
(iii) (Outer regularity) One has for every Borel measureable set .
Definition. (Haar measure) Let be a -compact locally Hausdorff group. A Radon measure is left-invariant (resp. right-invariant) if one has (resp. latex g\in G$ and Borel measureable sets . A left-invariant Haar measure is a nonzero Radon measure which is left-invariant.
Theorem. (Riesz representation theorem). Let be a -compact locally compact Hausdorff space. Then to every liner functional whihc is nonnegative (thus whenever ), one can associate a unique Radon measure such that for all . Conversely, for each Radon measure , the functional is a nonnegative linera functional on .