I'm looking for recommendations for books (or lecture notes) that develop measure theory in sufficient detail to state and prove the Riesz representation theorem (which is the characterization of the topological duals of linear spaces of continuous functions on a completely regular topological space). In particular, I'd like to see the cases of $C(X)$ and $C_K(X)$ of all continuous functions and all compactly supported continuous functions for non-compact $X$, with the usual Fréchet and locally convex topologies, covered as well as the more common case of compact $X$.
I know that this is basically in Dunford & Schwartz, but I always find it helpful to have multiple references at hand.
(24 Aug 2012) Update: There have been a number of helpful recommendations, however they are not all explicit about which version of the theorem is covered there. I've not yet had time to check them all, so let me collect here what I know so far, by theorem strength. I'm not aiming for a complete list. But it seems useful to have some list, since the stronger versions of the theorem don't appear to be so well known.
- Up to $C_K(X)$, locally convex topology. Many texts.
- Halmos, Measure Theory.
- Rudin, Real and Complex Analysis.
- Folland, Real Analysis.
- Aliprantis, Border, Infinite Dimensional Analysis.
- Fremlin, Topological Riesz Spaces and Measure Theory.
- Up to $C(X)$, Fréchet topology.
- Dunford, Schwartz, Linear Operators, Part 1.
- Berg, Christensen, Ressel, Harmonic Analysis on Semigroups.
A short and self contained treatment in the first two chapters.
- Up to $C(X)$, vector lattice from cone of positive units, order dual (since not a topological vector space).
To be honest, I'm not sure what the statement of the theorem is in this case, or what the standard associated to it. However, there there appear to be some results for this case.- König, Measure and Integration. Though, hard to interpret without an in depth reading.
Please leave a comment or an answer if you know where to place other references in this list.
Best Answer
May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) in the case where the underlying topological space is the real line and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).