A regular Borel measure 
on a Hausdorff space, in the
convention used here, is a Borel measure that can
be approximated from outside by open sets and from inside
by compact sets. For every Borel
set 
,
 |
and
 |
These properties are called outer regularity and inner regularity, respectively (Rudin 1987, p. 47). A complex measure is regular when its total variation is a regular Borel measure (Rudin 1987, p. 130).
A Borel regular outer measure uses a different regularity condition, requiring Borel sets
of equal outer measure around arbitrary subsets.
The 
-dimensional
Hausdorff outer measure on 
has this property (Falconer 1985, p. 8).
