Integral representation of linear functionals on function spaces

Abstract

Let $A$ be a vector space of real valued functions on a non-empty set $X$ and $L:A\to \mathbb{R}$ a linear functional. Given a $\sigma$-algebra $\mathcal{A}$, of subsets of $X$, we present a necessary condition for $L$ to be representable as an integral with respect to a measure $\mu$ on $X$ such that elements of $\mathcal{A}$ are $\mu$-measurable. This general result then is applied to the case where $X$ carries a topological structure and $A$ is a family of continuous functions and naturally $\mathcal{A}$ is the Borel structure of $X$. As an application, short solutions for the full and truncated $K$-moment problem are presented. An analogue of Riesz-Markov-Kakutani representation theorem is given where $C_c(X)$ is replaced with whole $C(X)$. Then we consider the case where $A$ only consists of bounded functions and hence is equipped with $\sup$-norm.