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Transactions of the American Mathematical Society
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0385067-1
PII: S 0002-9947(1975)0385067-1
Article copyright: © Copyright 1975 American Mathematical Society