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.
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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.
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].
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 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).
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].)
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].
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© 1980 Springer Science+Business Media New York
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Cohn, D.L. (1980). Measures on Locally Compact Spaces. In: Measure Theory. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-0399-0_7
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DOI: https://doi.org/10.1007/978-1-4899-0399-0_7
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