Chapter XI. Integration on Locally Compact Spaces

Abstract

This chapter deals with the special features of measure theory when the setting is a locally compact Hausdorff space and when the measurable sets are the Borel sets, those generated by the compact sets.

Sections 1–2 establish the basic theorem, the Riesz Representation Theorem, which says that any positive linear functional on the space $C_{\mathrm{com}}(X)$ of continuous scalar-valued functions of compact support on the underlying space $X$ is given by integration with respect to a unique Borel measure having a property called regularity. The steps in the construction of the measure run completely parallel to those for Lebesgue measure if one regards the geometric information about lengths of intervals as being encoded in the Riemann integral. The Extension Theorem of Chapter V is the main technical tool.

Section 3 studies more closely the nature of regularity of Borel measures. One direct generalization of a Euclidean theorem is that the space of continuous functions of compact support in an open set is dense in every $L^p$ space on that open set for $1\leq p \lt \infty$. A new result is the Helly–Bray Theorem—that any sequence of Borel measures of bounded total measure in a locally compact separable metric space has a weak-star convergent subsequence whose limit is a Borel measure.

Section 4 regards $C_{\mathrm{com}}(X)$ as a normed linear space under the supremum norm and identifies the space of continuous linear functionals, with its norm, as a space of signed or complex Borel measures with a regularity property, the norm being the total-variation norm for the signed or complex Borel measure.