nLab
Radon measure

Definition

If $X$ is a locally compact Hausdorff topological space, a Radon measure on $X$ is a Borel measure? on $X$ that is

• finite on all compact subsets,

• outer regular (i.e. can be approximated from outside by measure on the open sets) on all Borel sets, and

• inner regular (i.e. can be approximated from inside by a measure on compact sets) on open sets.

Properties

If a Radon measure is $\sigma$-finite then it is regular (i.e. both inner and outer regular) on all Borel subsets. Left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under left (right) multiplications by elements in the group.

Literature and related entries

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995