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.*