# Representation of a linear functional

We review results of Haviland and Riesz on the reperesentation of a linear functional.

Definition 1 Let ${X}$ be a subset of ${\mathbb{R}^n}$ and ${C(X)}$ be algebra of continuous functions on ${X}$. A positive linear functional on ${C(X)}$ is a linear functional ${L}$ with ${L(f) \ge 0}$ for all ${f \in C(X)}$ such that ${f(a) \ge 0, \forall a \in X}$.

We recall Haviland’s result in \cite{Marshall2} (also see \cite{Ha1, Ha2}), with ${\mathbb{R}[x_1, \dots, x_n]}$ denotes the ring of real multivariable polynomials:

Theorem 2 (Haviland) For a linear functional ${L: \mathbb{R}[x_1, \dots, x_n]}$ and closed set ${K}$ in ${\mathbb{R}^n}$, the following are equivalent:

1. ${L}$ comes from a Borel measure on ${K}$, i.e., ${\exists}$ a Borel measure ${\mu}$ on ${K}$ such that, ${\forall f \in \mathbb{R}[x_1, \dots, x_n], L(f) = \int f d\mu.}$
2. ${L(f) \ge 0}$ holds for all ${f \in \mathbb{R}[x_1, \dots, x_n]}$ such that ${f \ge 0}$ on ${K}$.

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 ${X}$ be a locally compact Hausdorff space and let ${L: C_c(X) \rightarrow \mathbb{R}}$ be a positive linear functional. Then there exists a unique Borel measure ${\mu}$ on ${X}$ such that

$\displaystyle L(f) = \int f d\mu, \forall f \in C_c(X).$

${C_c(X)}$ is the algebra of continuous functions with compact support.