Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Interchanging iterated integration


Author: Lawrence Lessner
Journal: Proc. Amer. Math. Soc. 68 (1978), 295-299
MSC: Primary 28A35
MathSciNet review: 0473134
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Abstract: If $ k(x,y)$ is a measurable, real valued, finite a.e. function on $ X \times Y$, then necessary and sufficient conditions are given for the two iterated Lebesgue integrals of $ k(x,y)$ to be equal and finite by employing Saks' theorem on the convergence of a sequence of finite measures and the Vitali convergence theorem. The conditions, more general than those of either Fubini's or Tonelli's theorems in this case, are applied to an example of a nonintegrable function to show that its iterated integrals are in fact equal and finite.


References [Enhancements On Off] (What's this?)

  • [1] Adriaan Cornelis Zaanen, Integration, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1967. Completely revised edition of An introduction to the theory of integration. MR 0222234
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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0473134-1
Keywords: Iterated integrals, Saks' theorem, Fubini-Tonelli's theorem, Vitali convergence theorem
Article copyright: © Copyright 1978 American Mathematical Society