# Measure theory treatment geared toward the Riesz representation theorem

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.

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Comment only, since I am not sure: is the account in Rudin's RCA general enough for your purposes? –  Yemon Choi Aug 21 '12 at 9:13
Don't know. Will check... That's why I'm asking. :-) –  Igor Khavkine Aug 21 '12 at 9:30
Hang on, what is the RRT for C(X) when X is non-compact? –  Yemon Choi Aug 21 '12 at 10:09
I think you can find these results in Aliprantis & Border, Infinite Dimensional Analysis (3rd ed). The book has a whole chapter on Riesz representation theorem. A hard to read book that probbly contains everything there is to know is Fremlin, Topological Riesz Spaces and Measure Theory. –  Michael Greinecker Aug 21 '12 at 10:30
@Yemon, according to Dunford & Schwartz, $C^*(X)$ is the space of "regular, bounded, (finitely) additive" set functions. I'm still trying to sort out what this means and how it relates to the topology on $C(X)$. –  Igor Khavkine Aug 21 '12 at 10:45

Have you looked at G.B. Folland's book "Real analysis, Modern techniques and their applications"? Chapter 7 of this books covers this topic. I have taught it once and I totally recommend it.

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The Notes after the chapter dealing with the Riesz representation theorem in Folland was quite helpful, being somewhat more up to date than the similar part of Dunford & Schwartz. Chasing down some of the references proved to be helpful. –  Igor Khavkine Aug 21 '12 at 23:26

Rudin's Real and Complex Analysis proves the theorem for the following two cases where $X$ is a locally compact Hausdorff space. :

• For linear functionals on the space $C_c(X)$, the space of all continuous compactly supported functions. (Theorem 2.14)

• For linear functionals on the space $C_0(X)$, the space of all continuous functions vanishing at infinity. (Theorem 6.19)

Since the second is the most general form of the theorem I know, surely this will suit your purposes?

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Thanks for looking checking! BTW, I just looked up Halmos, and he also seems to treat only the $C_K(X)$ case. –  Igor Khavkine Aug 21 '12 at 9:58
@Igor Please see edit. Rudin does treat a more general case. –  LeBlanc Aug 21 '12 at 10:08
I thought C_c(X) is more general than C_0(X), since it's a subalgebra? –  Yemon Choi Aug 21 '12 at 10:39
I see. In my understanding, treating $C_0(X)$ is the same as treating $C(\hat{X})$, where $\hat{X}$ is the one-point compactification. So it seems that Rudin only goes as far as treating $C(X)$ for compact $X$. –  Igor Khavkine Aug 21 '12 at 10:43
@LeBlanc: I meant that the RRT is more general for $C_c(X)$ because it characterizes functionals on a smaller algebra; it is not a priori clear that the functional on $C_c(X)$ extend to functionals on $C_0(X)$ without further assumptions on $X$. –  Yemon Choi Aug 22 '12 at 0:41

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, ...).

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Thanks for this interesting contribution! Although it's so information dense that it's hard to absorb in one reading. A quick question though. If, as you say, Buck worked with the dual of $C^b(X)$, is that not the same as working with the dual of $C(\beta X)$, where $\beta$ denotes Stone-Cech compactification? Or does his special choice of topology on $C^b(X)$ makes its topological dual identical with that of $C(X)$, with respect to some reasonable topology on the latter? –  Igor Khavkine Nov 10 '12 at 22:08
Buck uses weighted seminorms, i.e., he multiplies bounded, continuous functions by ones which vanish at infinity, then takes the supremum norm. This has the effect of cutting out the parts of the measures which live on the boundary. In the non locally compact case one mixes (in the sense of the polish school) the norm and the topology of compact convergence. It follows from the general theory that the new dual is the closure of that of the latter in the former, i.e., one gets those measures on the compactification which are approximable by ones with compact support and this is just right. –  jbc Nov 11 '12 at 6:47
@jbc, could you please share the definition of a $k_R$ space? I tried to find it in the literature but couldn't. –  Tom LaGatta Mar 10 at 21:38

V.I. Bogachev, Measure Theory, Vol. 2, Springer 2007 (Chapter 7)

D.H. Fremlin, Measure Theory, Vol. 4 (freely available, see www.essex.ac.uk/maths/people/fremlin/mt.htm)

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www.imsc.res.in/~sunder/rrt1.pdf

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Only treats up to $C_K(X)$, I think. –  Igor Khavkine Aug 25 '12 at 0:40