Measures and topology

  • Charalambos D. Aliprantis
  • Kim C. Border
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 4)


Chapter 8 dealt with measures and charges defined on abstract semirings or σ-algebras of sets. In applications there is often a natural topological or metric structure on the underlying measure space. By combining topological and set theoretic notions it is possible to develop a richer and more useful theory. Some of these connections between measure theory and topology are discussed in this chapter.


Borel Measure Polish Space Riesz Space Metrizable Space Countable Union 
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  1. 1.
    Be warned that there are several slightly different definitions of the Borel sets and Baire sets in common use. For instance, Kuratowski [150] defines the Borel sets to be the members of the smallest collection of sets including the closed sets that is closed under countable unions and countable intersections. For metric spaces this definition is equivalent to ours by Corollary 11.9. (Interestingly, in [149] he uses the same definition we do.) Halmos [103] defines the Borel sets of a locally compact Hausdorff space to be the members of the smallest σ-ring containing every compact set. This differs significantly from our definition—on an uncountable discrete space, only countable sets are Borel sets under this definition. For σ-compact spaces the two definitions agree. As for the Baire sets, Semadeni [209] defines the Baire sets to be the members of the smallest σ-algebra for which all members of C(X), the space of all continuous real functions, are measurable. (But he only considers compact spaces.) Halmos [103] defines the Baire sets to be the members of the σ-ring generated by the nonempty compact gδ’s. See Royden [200, Section 13.1, pp. 331–334] for an extended discussion of these various definitions. For locally compact separable metrizable spaces all these definitions agree.Google Scholar
  2. 2.
    If we take the Baire sets to be the members of the σ-ring generated by the compact G δ-sets, then the hypothesis of second countability may be dropped.Google Scholar
  3. 3.
    Recall that in Chapter 8 we required a Borel measure to assign finite measure to every compact set. Most authors make this definition. However, for the purposes of thisGoogle Scholar
  4. 4.
    This terminology is used by most, but not all, texts. Notably, Parthasarathy [187] uses “regular” to mean what we call outer regular. Most authors use “inner regular” to mean what we call “tight.” For compact Hausdorff spaces, there is no difference. Our use of the term “tight” agrees with the usage of the term by Billingsley [33]. The term “normal measure” is our own invention, and we find it useful.Google Scholar
  5. 5.
    Many authors do not require condition (2) as part of the definition of support. The support of a measure is often defined by supp μ = (∪ {V : V open and μ(V) = 0})C. By this definition, every measure has a (closed) support, but the support may not satisfy condition (2). See Example 11.26.Google Scholar
  6. 6.
    A point mass is sometimes called a Dirac measure or an evaluation. Google Scholar
  7. 7.
    Do not confuse this with ca r (Baire), the regular Baire measures. However if X is σ-compact and locally compact, then every regular Baire measure extends to a unique regular Borel measure. That is, ca r (Baire) = ca R(Baire). See, e.g., H. L. Royden [200, Corollary 12, p. 340, and Theorem 22, p.349].Google Scholar
  8. 8.
    There is one more interesting equivalent condition. Let C denote the convex set of all positive operators from C(X) into C(Y) that carry 1X onto 1Y—the set C is a convex subset of the vector space of all continuous operators from C(X) into C(Y). Then an operator T ∈ C is a lattice homomorphism if and only if T is an extreme point of C; for a proof see [205, Theorem 9.1, p. 195].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Charalambos D. Aliprantis
    • 1
  • Kim C. Border
    • 2
  1. 1.Department of MathematicsIndiana University-Purdue University Indianapolis (IUPUI)IndianapolisUSA
  2. 2.Division of the Humanities and Social SciencesCALTECHPasadenaUSA

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