The Riesz-Markov representation theorem states the following: if $X$ is a locally compact Hausdorff (LCH) space and $\ell: C_c(X) \to \mathbb{R}$ is a positive linear functional, then there is a unique Radon measure $\mu$ on $X$ such that $\ell(f) = \int_X f \, d\mu$ for all $f \in C_c(X)$. Here $C_c(X)$ denotes the space of continuous real-valued functions on $X$ with compact support.

In the case of compact metrizable $X$, I have seen this theorem proved as a consequence of Choquet's theorem, which you can read about on Wikipedia. I am aware that there is a generalization known as the Choquet-Bishop-de Leeuw theorem, but am not familiar with it.

Is there a version of Choquet's theorem that implies Riesz-Markov in the LCH generality in which I stated it above?

Edit: In the introduction to his book "Lectures on Choquet's theorem", Phelps sketches the proof of Riesz-Markov for compact Hausdorff spaces. (Further edit: this is not what he does; see the answer.) I have not yet worked through the question of whether this argument can be made to work in the LCH case. Will update if it does work.

Sunder (in this note) gives a method of obtaining the general LCH result using the compact case essentially as a black box, except that he obtains it for the Baire, rather than the Borel, $\sigma$-algebra. So a narrower question: is there a more straightforward way of obtaining the locally compact Riesz-Markov theorem from the compact Hausdorff case, yielding a measure on Borel, not just Baire, sets?


I would love it if someone could prove me wrong, but I think it's unlikely to expect any argument of this kind.

First, in the introduction to his book, Phelps does not in fact sketch the proof of Riesz-Markov for compact Hausdorff spaces. That was a misreading on my part earlier. Rather, he cites the Riesz-Markov theorem to illustrate the phenomenon that the book will examine. It turns out later that in the compact metric case, you can in fact use Choquet's theorem to prove Riesz-Markov, but citing the compact Hausdorff result in the introduction was to some extent rosy advertising. (And very effective advertising! It got my attention and made me read the relevant chapters of the book!)

Second, for generalizing from the compact metric to the compact Hausdorff setting, the problem that not all Baire sets are Borel probably cannot be avoided. This seems to be related to the fact that a compact Hausdorff space is separable if and only if it is metrizable, which is one of the many very nice applications of Urysohn's metrization theorem.

Third, compactness is used in the Choquet theory proofs via the property that $C_0(X)$ is a unital real algebra if and only if $X$ is compact. Here $C_0(X)$ is the space of real-valued continuous functions that vanish at infinity, i.e. are uniformly arbitrarily small outside a given compact set. Thus $C_0(X) = C(X)$ when $X$ is compact, and for general locally compact $X$, $C_0(X)$ is the uniform closure of $C_c(X)$. So even if you had a Choquet theory argument that somehow gave you a Borel measure on a compact Hausdorff space, the proof wouldn't obviously adapt to the locally compact setting.

However (fourth?), there is some light left in the world. Every locally compact Hausdorff space $X$ is homeomorphic to a compact Hausdorff space with one limit point $*$ deleted, in the subspace topology, namely the one-point compactification $X^*$. The point $*$ is seen from $X$ as the point at infinity. Moreover, $C_0(X)$ can be identified with the linear subspace (indeed, ideal) $\langle * \rangle$ of $C_0(X^*)$ consisting of functions that vanish at $*$ (see e.g this question). Note that $C_0(X^*) \simeq \langle * \rangle \oplus \mathbb{R}$. Let $\varphi$ be your most personally treasured positive linear functional on $\langle * \rangle$, and extend $\varphi$ to all of $C_0(X^*)$ via the M. Riesz extension theorem. We get a measure $\mu$ on $C_0(X^*)$, which restricts to $C_0(X)$ and clearly integrates everything in the way you'd like. (See also a more technical related question.)

The upshot is that you can indeed derive the locally compact from the compact result. For the additional cost of the locally compact result, over the cost in the compact one, two methods of payment are accepted: the one in e.g. Diestel and Spalsbury's The joys of Haar measure, using some compact neighbourhood book-keeping, or the above argument using a couple of facts about compactification and extension that need to be proved to keep things self-contained.


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