# A Variant Version of Riesz Representation Theorem

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 $$C_{K}$$.

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