nLab
Radon measure

Contents

Definition

If XX is a locally compact Hausdorff topological space, a Radon measure on XX is a Borel measure? on XX 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 σ\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.

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

Revised on April 18, 2012 16:38:41 by Toby Bartels (64.89.53.233)