16 - C(K) Spaces III

Summary

In this chapter we present Garling's proof [53] of the Riesz representation theorem for the dual of C(K), K compact Hausdorff. This theorem goes by a variety of names: The Riesz–Markov theorem, the Riesz–Kakutani theorem, and others. The version that we'll prove states:

Theorem 16.1.Let K be a compact Hausdorff space, and let T be a positive linear functional on C(K). Then there exists a unique positive Baire measure μ on K such that T (f) = ∫_K f d μ for every f ∈ C(K).

As we pointed out in the last chapter, our approach will be to first prove the theorem for l_∞ spaces. To this end, we will need to know a bit more about the Stone–Čech compactification of a discrete space and a bit more measure theory. First the topology.

The Stone–Čech Compactification of a Discrete Space

A topological space is said to be extremally disconnected, or Stonean, if the closure of every open set is again open. Obviously, discrete spaces are extremally disconnected. Less mundane examples can be manufactured from this starting point:

Lemma 16.2.If D is a discrete space, then βD is extremally disconnected.

Proof. Let U be open in βD, and let A = U ∩ D. Then A is dense in U since U is open, and so cl_βD A = cl_βDU. Now we just check that cl_βDA is also open. The characteristic function χ_A : D → {0, 1} (a continuous function on D!) extends continuously to some f : βD → {0, 1}. Thus, by continuity, cl_βDA = f^-1({1}) is open.