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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integration of functions with values in locally convex Suslin spaces

Author: G. Erik F. Thomas
Journal: Trans. Amer. Math. Soc. 212 (1975), 61-81
MSC: Primary 28A45
MathSciNet review: 0385067
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Abstract: The main purpose of the paper is to give some easily applicable criteria for summability of vector valued functions with respect to scalar measures. One of these is the following: If E is a quasi-complete locally convex Suslin space (e.g. a separable Banach or Fréchet space), $ H \subset E'$ is any total subset, and f is an E-valued function which is Pettis summable relative to the ultra weak topology $ \sigma (E,H)$. f is actually Pettis summable for the given topology. (Thus any E-valued function for which the integrals over measurable subsets can be reasonably defined as elements of E is Pettis summable.) A class of ``totally summable'' functions, generalising the Bochner integrable functions, is introduced. For these Fubini's theorem, in the case of a product measure, and the differentiation theorem, in the case of Lebesgue measure, are valid. It is shown that weakly summable functions with values in the spaces $ D,E,S,D',E',S'$, and other conuclear spaces, are ipso facto totally summable.

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Article copyright: © Copyright 1975 American Mathematical Society