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An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions

Author

Listed:
  • Clara Carlota

    (Departamento de Matemática, CIMA, Universidade de Évora, 7000-671 Évora, Portugal
    These authors contributed equally to this work.)

  • António Ornelas

    (Departamento de Matemática, CIMA, Universidade de Évora, 7000-671 Évora, Portugal
    These authors contributed equally to this work.)

Abstract

In this paper, our main aim is to present a reasonable extension of the 1-dim Lebesgue integral of the product of two functions, in case this Lebesgue integral does not exist (i.e., the integrals of its negative and positive parts are both ∞ ). This extension works fine quite generally, as shown by several examples, and it is based on general hypotheses guaranteeing the sign of the integral (in the sense of being necessarily <0 or =0 or else >0), without computing its actual value. For this purpose, our method provides much more precise results than the Lebesgue–Stieltjes integration by parts.

Suggested Citation

  • Clara Carlota & António Ornelas, 2023. "An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions," Mathematics, MDPI, vol. 11(15), pages 1-16, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3341-:d:1206535
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