I have a positive linear functional $h$ defined on a set of Lesbesgue-measurable functions of "moderate growth" on $\mathbb{R}^2$–call this set $MG(\mathbb{R}^2)$. (A function $f$ is positive if $f(x)\ge 0$ for all $x \in \mathbb{R}^2$.) I want to ensure that there is a unique measure $\mu$ such that $$ h(f) = \int f\, \mathrm{d} \mu \quad\text{for all} \quad f \in MG(\mathbb{R}^2).$$
Let me describe a line of attack. Let $C_c(\mathbb{R}^2)$ be the set of continuous, compactly supported functions on $\mathbb{R}^2$. The Reisz–Markov–Kakutani representation theorem ensures that there exists a unique measure $\mu$ on $\mathbb{R}^2$ such that
$$ h(f) = \int f\, \mathrm{d} \mu \quad\text{for all} \quad f \in C_c(\mathbb{R}^2)\subset MG(\mathbb{R}^2).$$
To what extent can I extend this representation to all of $MG(\mathbb{R}^2)$? Is there some general result that ensures positive linear functionals have a unique integral representation for a larger class than just $C_c(\mathbb{R}^2)$? If you have a specific reference, that would be immensely helpful.
Finally, if it helps to have a concrete example in mind, I can take $MG(\mathbb{R}^2)$ to be the set of all measurable functions that grow no faster than $O(\exp(x^2+y^2))$ as $x^2+y^2\to \infty$.
I would not expect that the representation
$$ h(f) = \int_{\mathbb R^2} f\, \mathrm{d} \mu \quad\text{for all} \quad f \in C_c(\mathbb{R}^2)\subset MG(\mathbb{R}^2)$$
can be extended to a "much larger" subspace (you can extend it e.g. to $C_0(\mathbb R^2)$) without any additional conditions on $h$. The reason is that $h$ may be nontrivial, but equal to zero on $C_c(\mathbb R^2).$ You can construct such $h$ using M. Riesz extension theorem.
Let e.g. $$MG(\mathbb R^2)=\{f\mid f \mbox{ measurable and }|f(x,y)e^{-x^2-y^2}|<C\ \mbox{for some}\ C\}$$ with $||f||=\sup_{x,y}|f(x,y)|e^{-x^2-y^2}.$ Define $h$ on the subspace
$$ F=\{f\in C(\mathbb R^2)\mid \exists\lim_{x^2+y^2\to\infty} f(x,y)e^{-x^2-y^2}\} $$ as $$h(f)=\lim_{x^2+y^2\to\infty} f(x,y)e^{-x^2-y^2}$$ and extend it to a positive functional on $MG(\mathbb R^2).$ Then $h\equiv 0$ on $C_c(\mathbb R^2),$ hence $\mu\equiv 0$ ($\mu$ as in Riesz-Markov-Kakutani theorem).