Abstract
Chapter 7 is devoted to the Riesz representation theorem and related results. The first section (Section 7.1) contains some basic facts about locally compact Hausdorff spaces, the spaces that provide the natural setting for the Riesz representation theorem, while the second section (Section 7.2) gives a proof of the Riesz representation theorem. The next two sections (Sections 7.3 and 7.4) contain some useful and relatively basic related material. The results of Sections 7.5 and 7.6 are needed for dealing with large locally compact Hausdorff spaces; for relatively small locally compact Hausdorff spaces (those that have a countable base), very few of the results in those two sections are needed.
The Daniell-Stone integral gives another way to deal with integration on locally compact Hausdorff spaces. Section 7.7 contains a result due to Kindler that summarizes the relationship of the Daniell-Stone integral to measure theory. The general Daniell-Stone setup is outlined in the exercises at the end of Section 7.7.
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Notes
- 1.
Recall that a linear functional I on a vector space of functions is positive if I(f) ≥ 0 holds for each nonnegative function f in the domain of I.
- 2.
A topological space is discrete (or has the discrete topology) if each of its subsets is open.
- 3.
See Theorem D.20.
- 4.
The assumption that A is compact simplifies the proof, but is not actually necessary.
- 5.
Note that μ is a positive measure, since its specification has no modifier such as “signed” or “complex.”
- 6.
Note that \({I}^{{\ast}}(\vert f\vert ) = I(\vert f\vert ) < +\infty \) holds for each f in \(\mathcal{K}(X)\) and hence that \(\mathcal{K}(X)\) is included in \({\mathcal{F}}^{1}\).
- 7.
Actually, he usually calls his positive linear functional μ, and he writes μ(f) and ∫f dμ, rather than I(f) and ∫f dI; such notation will not be used in this book, since we have been using μ to denote a measure.
- 8.
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Cohn, D.L. (2013). Measures on Locally Compact Spaces. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_7
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