Radon Measures

Summary

7.1 For the convenience of the reader, we review some properties of locally compact HausdorfF spaces, such as Urysohn’s Lemma.

7.2 Let X be a locally compact Hausdorff space. A linear form μ on the space of complex-valued continuous functions with compact support H is a Radon measure if, for every compact K ⊂ X, the restriction of μ to the space of continuous functions with support in K is continuous. Every positive linear form on H is a Radon measure (Theorem 7.2.1) and every Radon measure is a Daniell measure. Then we define the upper integral with respect to a Radon measure and give results analogous to those obtained in Chapter 3.

7.3 The main result of this section (Theorem 7.3.2) is the following. The upper envelope f of an upwardly directed set, H, of integrable, lower semicontinuous functions such that sup_g∈H∫gdVμ< ∞ is itself integrable and g converges to f in the mean along the filter of sections of H. Also, a Radon measure is regular in the following sense: if A is measurable and moderate, then, given ε > 0, there is an open set U and a countable union of compact sets F such that Vμ^* (U — F) ≤ ε.

7.4 In this section we introduce the notion of Lusin measurable mappings. Intuitively these functions are “almost continuous” on any compact set. A function from X into a metrizable space is Lusin measurable if and only if it is measurable.

7.5 Let x_o ∈ X. The measure ε _{x o}, defined by f ↦ f(x_o) for f continuous with compact support, shows that a Radon measure may have some atoms. However, any atom is essentially a point (Proposition 7.5.1). Counting measures provide us with examples of atomic Radon measures.

7.6 One of the goals of measure theory was to improve the Riemann integral. The reader might be happy to know that what we have done so far has substantially improved Riemann’s integral. In this section we prove, in particular, that a bounded function with compact support is Riemann integrable if and only if its set of discontinuity is Lebesgue negligible.

7.7 Various types of convergence, such as vague, weak, and narrow convergence in spaces of measures are studied.

7.8 We continue our investigation of convergence in spaces of measures by looking at tight subsets of M¹. Tightness may be viewed as a relative compactness condition.