A Borel regular outer measure on a topological space 
is an outer measure 
for which every Borel set
is measurable in the sense that
 |
for every subset 
and Borel set 
.
Moreover, for every subset 
there is a Borel set 
with 
.
Restriction to the Borel sets gives a Borel
measure. The 
-dimensional Hausdorff outer
measure on 
is an example (Falconer 1985, pp. 5-8).
This Borel-envelope condition is different from the inner and outer approximations defining a regular Borel measure. It asks for a Borel set containing an arbitrary subset, rather than for open sets containing a Borel set.
