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


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.


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


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© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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