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The following is an excerpt from David R. Adams, Function Spaces and Potential Theory, page 203. For the sake of convenience, I will simplify the notations which will be slightly different than the original excerpt.

Assume that the underlying space is Euclidean $\mathbb{R}$, and $X$ is a normed space that includes $C_{0}(\mathbb{R})$ and $C(K)$, $K$ is an arbitrary compact set in $\mathbb{R}$, where $C(K)$ is the set of all continuous functions on $K$ and $C_{0}(\mathbb{R})$ is the set of all continuous functions with compact support in $\mathbb{R}$. Suppose further that we are given a bounded linear operator $L:X\rightarrow\mathbb{R}$ such that for each compact set $K$, there is a constant $C_{K}$ depending on $K$ such that \begin{align*} |L(f)|\leq C_{K}\|f\|_{L^{\infty}},~~~~f\in C(K). \end{align*} Adams claimed that by Riesz Representation Theorem one can find some Radon measure $\mu$ such that \begin{align*} L(f)=\int_{\mathbb{R}}fd\mu,~~~~f\in C_{0}(\mathbb{R}). \end{align*}

I don't see how it is derived, but I have some speculation about it. First of all, since $L\in(C(K))^{\ast}$, by the usual Riesz Represenation Theorem one can find some Radon measure $\mu_{K}$ such that \begin{align*} L(f)=\int_{K}fd\mu_{K},~~~~f\in C(K). \end{align*} Consider an exhaustion of $\mathbb{R}$, say, $\mathbb{R}=\bigcup_{n\geq 1}[-n,n]:=\bigcup_{n\geq 1}K_{n}$ and the realizing measure is denoted by $\mu_{n}$. I suspect that \begin{align*} \mu_{n+1}(S)=\mu_{n}(S) \end{align*} for any Borel set $S\subseteq K_{n}$, then I think one shall consider the "measure" $\mu$ defined by \begin{align*} \mu(K)=\lim_{n\rightarrow\infty}\mu_{n}(K) \end{align*} for any compact set $K$ in $\mathbb{R}$, as the sequence $\{\mu_{n}(K)\}_{n}$ is eventually constant. Then one shall define $\mu(U)=\sup\{\mu(K): K\subseteq U, K~\text{compact}\}$ for an open set $U$ and then $\mu(E)=\inf\{\mu(U): U\supseteq E, U~\text{open}\}$. But this may have some flaw about the definition, whether the definition regarding of the infimum one coincides with the usual $\mu(K)$ for a compact set $K$ is in question. With this "measure", I think this is the one that Adams noted as in the integral representation for $L$ on $C_{0}(\mathbb{R})$.

Note that the functions discussed above are all assumed in real-valued, but it seems to me that Adams didn't emphasize that the underlying functions must be real-valued, I wonder what would happen if complex-valued functions are taken account.

Also note that if $L$ is a nonnegative measure, then some version of Riesz Representation Theorem will do the job, but here we don't assume the nonnegativity.

I also think of the constants $C_{K}$ that depend on $K$ will cause a problem of looking for such a measure $\mu$ or not. Perhaps we need a uniform constant $C$ rather than those $C_{K}$.

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  • $\begingroup$ You should state what the domain of $L$ is at the beginning. $\endgroup$ – zhw. Aug 14 at 16:54

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