Any Radon measure is
tight
(also called
inner regular):
For any Borel subset
of
one has
If
is countably generated,
is a Radon space if and only if it
is Borel isomorphic to a universally measurable subset of
(or any other uncountable compact metrizable space). In particular, any
polish space,
or more generally
Suslin space
in the sense of Bourbaki, is Radon.
One can also define non-finite (non-negative) Radon measures;
they are tight and take finite values on compact subsets. If
has a countable basis, they are
-finite.
Following
N. Bourbaki
(and ideas going back to
W.H. Young
and
Ch. de la Vallée-Poussin),
a (non-negative)
Radon measure on, say, a locally compact space
is a (non-negative) continuous linear functional on the space
of continuous functions with compact support endowed with
its natural inductive topology. One can prove with the help of the
Riesz–Markov theorem
(which deals with the case
compact) that any non-negative and bounded Radon measure
in this sense is the restriction to
of the integral with respect to a unique (finite)
Radon measure as defined in the article above; the converse is true and trivial.