## Abstract

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.

## Keywords

Open Subset Compact Subset Compact Space Borel Subset Compact Hausdorff Space
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## Notes

- The historical notes in Chapter III of Hewitt and Ross [41] 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 [38], Hewitt and Stromberg [42], Rudin [75], or Hewitt and Ross [41] useful. He (or she) would also do well to look up the paper of Kakutani [48].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 [41], Hewitt and Stromberg [42], and Rudin [75], but with that given by Halmos [38] only when*X*is σ-compact. The definition given in Exercise 7.2.8 for the σ-algebra ℬ_{0}(*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 ℬ_{0}(*X*) that can reasonably be applied to an arbitrary locally compact Hausdorff space*X*).Google Scholar - Bourbaki [11] and Hewitt and Ross [41] 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 [36] and Bledsoe and Wilks [8].)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 [58] and Royden [73].Google Scholar

## Copyright information

© Springer Science+Business Media New York 1980