In this chapter we shall be dealing with measures and integrals on locally compact Hausdorff spaces. This first section contains a summary of some of the necessary topological facts and constructions; the main development begins in Section 2.
KeywordsOpen Subset Compact Subset Compact Space Borel Subset Compact Hausdorff Space
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- The historical notes in Chapter III of Hewitt and Ross  contain a nice summary of the history of integration theory on locally compact Hausdorff spaces.Google Scholar
- The reader who wants to see another elementary treatment of integration on locally compact Hausdorff spaces might find Halmos , Hewitt and Stromberg , Rudin , or Hewitt and Ross  useful. He (or she) would also do well to look up the paper of Kakutani .Google Scholar
- The definition given here for the σ-algebra ℬ(X) of Borel subsets of X agrees with that given by Hewitt and Ross , Hewitt and Stromberg , and Rudin , but with that given by Halmos  only when X is σ-compact. The definition given in Exercise 7.2.8 for the σ-algebra ℬ0(X) of Baire subsets of a compact Hausdorff space X is a special case of that given by Halmos (Halmos considers σ-rings, in addition to σ-algebras, and so is able to give a definition of ℬ0 (X) that can reasonably be applied to an arbitrary locally compact Hausdorff space X).Google Scholar
- Bourbaki  and Hewitt and Ross  deal with the μ*-, υ*-, and (μ, × υ)*-measurable sets, rather than with the Borel sets, when considering product measures. Proposition 7.6.5, Corollary 7.6.6, and Theorem 7.6.7 were suggested by deLeeuw [241. (See also Godfrey and Sion  and Bledsoe and Wilks .)Google Scholar
- The Daniell treatment of integration theory (see the notes at the end of Chapter 2) can also be used to prove a version of the Riesz representation theorem; this is done, for instance, by Loomis  and Royden .Google Scholar