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Integrals and measures

In: Measure and Integration

Author

Listed:
  • Hari Bercovici

    (Indiana University, Department of Mathematics)

  • Arlen Brown

    (Indiana University, Department of Mathematics)

  • Carl Pearcy

    (Texas A&M University, Department of Mathematics)

Abstract

In the language of modern integration theory the term integral integral refers to a number of somewhat different concepts, arrived at through a variety of constructions and definitions. About the only thing that can be said about integration in reasonable generality is that an integral on a space X is a linear transformation linear transformation transformation -linear that is defined on a vector space vector space -of functions of functions on X and satisfies certain continuity continuity requirements. As regards the Lebesgue integral -Lebesgue integral, Lebesgue integral however, matters are in a much less chaotic state. Indeed, while a considerable number of different definitions and L: a Lebesgue integral constructions ℒ $$\mathcal{L}$$ : the domain of a Lebesgue integral can be found in the literature, there is unanimous agreement on what a Lebesgue integral is. We provide an axiomatic characterization.

Suggested Citation

  • Hari Bercovici & Arlen Brown & Carl Pearcy, 2016. "Integrals and measures," Springer Books, in: Measure and Integration, edition 1, chapter 0, pages 43-73, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-29046-1_3
    DOI: 10.1007/978-3-319-29046-1_3
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