A Radon measure on a Hausdorff space 
is a Borel measure such
that every point has a neighborhood
of finite measure and the
measure of each Borel set
is the supremum of the measures
of compact sets contained in it (Teschl 2014, p. 311).
The finite-neighborhood condition makes this a locally finite measure, and the approximation by compact sets
is inner regularity. A Radon measure need not have finite total measure.
For example, Lebesgue measure on 
is a Radon measure with infinite total measure.
A Radon measure of finite total measure is a regular Borel measure in the convention requiring both inner and outer regularity. Indeed,
approximate the complement set of a Borel
set 
from inside by a compact set 
. The open set 
then contains 
and approximates its measure from
above.
