Interchanging iterated integration
Abstract: If is a measurable, real valued, finite a.e. function on , then necessary and sufficient conditions are given for the two iterated Lebesgue integrals of 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.
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-  Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
-  N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1967.
-  Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
- A. C. Zaanen, Integration, North-Holland Publishing Company, Amsterdam, 1967. MR 0222234 (36:5286)
- P. R. Halmos, Measure theory, D. Van Nostrand Company, New York, 1950. MR 0033869 (11:504d)
- N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1967.
- A. L. Peressini, Ordered topological vector spaces, Harper and Row, New York, 1967. MR 0227731 (37:3315)
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Keywords: Iterated integrals, Saks' theorem, Fubini-Tonelli's theorem, Vitali convergence theorem
Article copyright: © Copyright 1978 American Mathematical Society