Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1978-0473134-1
MathSciNet review: 0473134
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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] A. C. Zaanen, Integration, North-Holland Publishing Company, Amsterdam, 1967. MR 0222234 (36:5286)
  • [2] P. R. Halmos, Measure theory, D. Van Nostrand Company, New York, 1950. MR 0033869 (11:504d)
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1967.
  • [4] A. L. Peressini, Ordered topological vector spaces, Harper and Row, New York, 1967. MR 0227731 (37:3315)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A35

Retrieve articles in all journals with MSC: 28A35


Additional Information

DOI: https://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

American Mathematical Society