We review results of Haviland and Riesz on the reperesentation of a linear functional.
Definition 1 Let be a subset of
and
be algebra of continuous functions on
. A positive linear functional on
is a linear functional
with
for all
such that
.
We recall Haviland’s result in \cite{Marshall2} (also see \cite{Ha1, Ha2}), with denotes the ring of real multivariable polynomials:
Theorem 2 (Haviland) For a linear functional and closed set
in
, the following are equivalent:
comes from a Borel measure on
, i.e.,
a Borel measure
on
such that,
holds for all
such that
on
.
In Haviland’s theorem, a positive linear functional extended from ring of real multivariable polynomials to larger subalgebra and this theorem can be derived as a consequence of the following Riesz Representation Theorem (see \cite[p. 77]{KS}):
Theorem 3 (Riesz Representation Theorem) Let be a locally compact Hausdorff space and let
be a positive linear functional. Then there exists a unique Borel measure
on
such that
is the algebra of continuous functions with compact support.