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.