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Integrals for functions with values in a partially ordered vector space.

A C M van Rooij1, W B van Zuijlen2

  • 11Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands.

Positivity
|April 10, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces extensions for integrating functions in partially ordered vector spaces. Applying these extensions to simple functions recovers the standard space of integrable functions.

Keywords:
Pettis integralBochner integralIntegralLateral extensionPartially ordered vector spaceRiesz spaceVertical extension

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Area of Science:

  • Real Analysis
  • Functional Analysis
  • Measure Theory

Background:

  • Integration theory is fundamental in real analysis.
  • Partially ordered vector spaces provide a framework for functions with order structures.
  • The space of integrable functions is a cornerstone in mathematical analysis.

Purpose of the Study:

  • To explore extensions of the space of integrable functions within partially ordered vector spaces.
  • To investigate novel integration concepts for functions with values in these spaces.

Main Methods:

  • Consideration of functions mapping measure spaces to partially ordered vector spaces.
  • Development of two distinct extension methodologies for the space of integrable functions.
  • Application of these extensions to real-valued simple functions.

Main Results:

  • The proposed extensions successfully generalize the concept of integration.
  • Applying the extensions to simple functions yields the classical Lebesgue integrable function space.
  • Demonstration of the equivalence between generalized and classical integration under specific conditions.

Conclusions:

  • The introduced extensions offer a robust framework for integration in partially ordered vector spaces.
  • This work bridges generalized integration concepts with classical Lebesgue integration.
  • The findings have implications for advanced mathematical analysis and related fields.