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Measure

  • Marek Capiński
  • Peter Ekkehard Kopp
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

The idea of a ‘negligible’ set relates to one of the limitations of the Riemann integral, as we saw in the previous chapter. Since the function f = 1 takes a non-zero value only on ℚ, and equals 1 there, the ‘area under its graph’ (if such makes sense) must be very closely linked to the ‘length’ of the set ℚ. This is why it turns out that we cannot integrate f in the Riemann sense: the sets ℚ and ℝ\ℚ are so different from intervals that it is not clear how we should measure their ‘lengths’ and it is clear that the ‘integral’ of f over [0, 1] should equal the ‘length’ of the set of rationals in [0, 1]. So how should we define this concept for more general sets?

Keywords

Lebesgue Measure Measure Space Pairwise Disjoint Countable Union Outer Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Limited 2004

Authors and Affiliations

  • Marek Capiński
    • 1
  • Peter Ekkehard Kopp
    • 2
  1. 1.Institute of MathematicsJagiellonian UniversityKrakówPoland
  2. 2.Department of MathematicsUniversity of HullHullUK

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