Measures

  • Donald L. Cohn
Chapter
Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

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

Sigma-algebra Borel set Measure Measurable space Outer measure Measurable set Lebesgue measure Cantor set Non-measurable set Complete measure Inner measure 

References

  1. 3.
    Bartle, R.G.: The Elements of Integration. Wiley, New York (1966)Google Scholar
  2. 6.
    Benedetto, J.J., Czaja, W.: Integration and Modern Analysis. Birkhäuser, Boston (2012)Google Scholar
  3. 7.
    Berberian, S.K.: Measure and Integration. Macmillan, New York (1965). Reprinted by AMS Chelsea Publishing, 2011Google Scholar
  4. 8.
    Billingsley, P.: Probability and Measure. Wiley, New York (1979)Google Scholar
  5. 14.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29. Academic, New York (1968). Reprinted by Dover, 2007Google Scholar
  6. 15.
    Bogachev, V.I.: Measure Theory, 2 vols. Springer, Berlin (2007)Google Scholar
  7. 23.
    Bruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis, 2nd edn. ClassicalRealAnalysis.com (2008)Google Scholar
  8. 40.
    Dudley, R.M.: Real Analysis and Probability, 2nd edn. Cambridge University Press, Cambridge (2002)Google Scholar
  9. 43.
    Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse. Die Grundlehren der mathematischen Wissenschaften, Band 108. Springer, Berlin (1961)Google Scholar
  10. 44.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  11. 45.
    Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley, New York (1999)Google Scholar
  12. 46.
    Fremlin, D.H.: Measure Theory, 5 vols. www.essex.ac.uk/maths/people/fremlin/mt.htm
  13. 48.
    Gelbaum, B.R., Olmsted, J.M.H.: Counterexamples in Analysis. Holden-Day, San Francisco (1964). Reprinted by Dover, 2003Google Scholar
  14. 54.
    Halmos, P.R.: Measure Theory. Van Nostrand, Princeton (1950). Reprinted by Springer, 1974Google Scholar
  15. 59.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1965)Google Scholar
  16. 75.
    Krantz, S.G., Parks, H.R.: Geometric Integration Theory. Birkhäuser, Boston (2008)Google Scholar
  17. 80.
    Lang, S.: Algebra. Addison-Wesley, Reading (1965)Google Scholar
  18. 89.
    Morgan, F.: Geometric Measure Theory: A Beginner’s Guide. Academic, San Diego (2000)Google Scholar
  19. 92.
    Munroe, M.E.: Measure and Integration, 2nd edn. Addison-Wesley, Reading (1971)Google Scholar
  20. 95.
    Pap, E. (ed.): Handbook of Measure Theory, 2 vols. North Holland (Elsevier), Amsterdam (2002)Google Scholar
  21. 100.
    Rogers, C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970)Google Scholar
  22. 102.
    Royden, H.L.: Real Analysis, 2nd edn. Macmillan, New York (1968)Google Scholar
  23. 105.
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)Google Scholar
  24. 110.
    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)Google Scholar
  25. 127.
    Wheeden, R.L., Zygmund, A.: Measure and Integral. Monographs and Textbooks in Pure and Applied Mathematics, vol. 43. Marcel Dekker, New York (1977)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Donald L. Cohn
    • 1
  1. 1.Department of Mathematics and Computer ScienceSuffolk UniversityBostonUSA

Personalised recommendations