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.
If a Radon measure is $\sigma$-finite then it is regular (i.e. both inner and outer regular) on all Borel subsets.
A Radon measure on a Hausdorff space is τ-additive. The converse is true on a compact Hausdorff space.
Radon probability measures on compact Hausdorff spaces form a monad: the Radon monad. Just as well, Radon probability measures of finite first moment on complete metric spaces give the Kantorovich monad.
(See also monads of probability, measures and valuations.)
Most measures of interest in geometry are Radon. For example
The Dirac measures.
The Lebesgue measure on the real line.
The measure associated to a volume form on a Riemannian manifold.
The 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.
V. Bogachev, Measure Theory, vol. 2 (2007).
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995
Last revised on October 22, 2019 at 16:00:27. See the history of this page for a list of all contributions to it.