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*.

- Halmos,
- 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.

- Dunford, Schwartz,
- 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.

- König,

Please leave a comment or an answer if you know where to place other references in this list.

isthe RRT for C(X) when X is non-compact? $\endgroup$ – Yemon Choi Aug 21 '12 at 10:09Infinite 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. $\endgroup$ – Michael Greinecker Aug 21 '12 at 10:30