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The Lebesgue Integral

  • Edwin Hewitt
  • Karl Stromberg

Abstract

Integration from one point of view is an averaging process for functions, and it is in this spirit that we will introduce and discuss integration. In applying an averaging process to a class \({\mathcal{F}}\) real- or complex-valued functions, a number I(f) is assigned to each \(f \in {\mathcal{F}}\). If I(f) is to be an average, then it should certainly satisfy the conditions
$$\matrix{{I(f + g) = I(f) + I(g),} \cr {I(\alpha f) = \alpha I(f)} \cr }$$
for f, \(g \in {\mathcal{F}}\) and α ∈R. A less essential but often desirable property for I is that I(f) ≧ 0 if f ≧ 0. In some cases these three properties suffice to identify the averaging process completely.

Keywords

Measure Space Compact Hausdorff Space Outer Measure Nondecreasing Sequence Finite Measure Space 
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 Berlin · Heidelberg 1965

Authors and Affiliations

  • Edwin Hewitt
    • 1
  • Karl Stromberg
    • 2
  1. 1.The University of WashingtonUSA
  2. 2.The University of OregonUSA

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