Let us recall the following data related to Borel space and measure theory:
sigma-algebra, or
-algebra;
the Borel algebra which is defined as the smallest
-algebra on the field of real numbers
generated by the open intervals of
;
Borel space
Consider a locally compact Hausdorff space
; a Borel measure is then defined as any measure
on the sigma-algebra of Borel sets, that is, the Borel
-algebra
defined on a locally compact Hausdorff space
;
When the Borel measure
is both inner and outer regular on all Borel sets, it is called a regular Borel measure;
Recall that a topological space
is
-compact if there exists a sequence
of compact subsets
of
such that :
Definition 0.1
Let
be a Borel space (with the
-algebra
of Borel sets of a topological space
), and let
be a measure on the space
. Then, such a measure is called a
-finite (Borel) measure
if there exists a sequence
with
for all
, such that