A measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume, etc. Measure theory is concerned with generalizing the notions of area on arbitrary sets of Euclidean spaces and notions of length of subsets of R. Essentially, it is a common ground for analysis of real functions and set theory.

A first course in measure theory will most likely devote time to many of the following topics:

  • Functions and integrals
  • Convergence
  • Signed and complex measures
  • Product measures
  • Differentiation
  • Measures on locally compact spaces
  • Polish spaces and analytic sets
  • Haar measure
  • Operations on measures
  • Extensions
  • Structural characteristics for set functions
  • Measurable functions on monotone measure spaces
  • Integration
  • Sugeno integrals
  • Pan-integrals
  • Choquet integrals
  • Upper and lower integrals
  • Constructing general measures
  • Fuzzification of generalized measures and the Choquet integral
  • Applications of generalized measure theory

A graduate level text on measure theory is available on Amazon.com. Students can find a large number of books on this topic on Google, and there are some measure theory tutorials available online.

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