Abstract
In the most common construction of the Lebesgue integral of a function, the definition of the integral assumes that one knows the sizes of subsets of the function's domain. In Chapter 1 we introduce measures, the basic tool for dealing with such sizes. The first two sections of the chapter are abstract (but elementary). Section 1.1 looks at sigma-algebras, the collections of sets whose sizes we measure, while Section 1.2 introduces measures themselves. The heart of the chapter is in the following two sections, where we look at some general techniques for constructing measures (Section 1.3) and at the basic properties of Lebesgue measure (Section 1.4). The chapter ends with Sections 1.5 and 1.6, which introduce some additional fundamental techniques for handling measures and sigma-algebras.
Keyword
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The terms field and σ-field are sometimes used in place of algebra and σ-algebra.
- 2.
See Chap. 8 for some interesting and useful sets that are not Borel sets.
- 3.
If in Example 1.2.1(a) the σ-algebra \(\mathcal{A}\) contains all the subsets of X, then μ is σ-finite if and only if X is at most countably infinite.
- 4.
See items A.12 and A.13 in Appendix A.
- 5.
For details, see Solovay [110].
- 6.
This means that B spans \(\mathbb{R}\) (i.e., that \(\mathbb{R}\) is the smallest linear subspace of \(\mathbb{R}\) that includes B) and that no proper subset of B spans \(\mathbb{R}\). The axiom of choice implies that such a set B exists; see, for example, Lang [80, Section 5 of Chapter III].
References
Bartle, R.G.: The Elements of Integration. Wiley, New York (1966)
Benedetto, J.J., Czaja, W.: Integration and Modern Analysis. Birkhäuser, Boston (2012)
Berberian, S.K.: Measure and Integration. Macmillan, New York (1965). Reprinted by AMS Chelsea Publishing, 2011
Billingsley, P.: Probability and Measure. Wiley, New York (1979)
Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29. Academic, New York (1968). Reprinted by Dover, 2007
Bogachev, V.I.: Measure Theory, 2 vols. Springer, Berlin (2007)
Bruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis, 2nd edn. ClassicalRealAnalysis.com (2008)
Dudley, R.M.: Real Analysis and Probability, 2nd edn. Cambridge University Press, Cambridge (2002)
Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse. Die Grundlehren der mathematischen Wissenschaften, Band 108. Springer, Berlin (1961)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)
Fremlin, D.H.: Measure Theory, 5 vols. www.essex.ac.uk/maths/people/fremlin/mt.htm
Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. Holden-Day, San Francisco (1964). Reprinted by Dover, 2003
Halmos, P.R.: Measure Theory. Van Nostrand, Princeton (1950). Reprinted by Springer, 1974
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1965)
Krantz, S.G., Parks, H.R.: Geometric Integration Theory. Birkhäuser, Boston (2008)
Lang, S.: Algebra. Addison-Wesley, Reading (1965)
Morgan, F.: Geometric Measure Theory: A Beginner’s Guide. Academic, San Diego (2000)
Munroe, M.E.: Measure and Integration, 2nd edn. Addison-Wesley, Reading (1971)
Pap, E. (ed.): Handbook of Measure Theory, 2 vols. North Holland (Elsevier), Amsterdam (2002)
Rogers, C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970)
Royden, H.L.: Real Analysis, 2nd edn. Macmillan, New York (1968)
Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)
Solovay, R.M.: A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2) 92, 1–56 (1970)
Wheeden, R.L., Zygmund, A.: Measure and Integral. Monographs and Textbooks in Pure and Applied Mathematics, vol. 43. Marcel Dekker, New York (1977)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Cohn, D.L. (2013). Measures. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6956-8_1
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4614-6955-1
Online ISBN: 978-1-4614-6956-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)