Integration of functions with values in locally convex Suslin spaces

Thomas, G. Erik F.

doi:10.1090/S0002-9947-1975-0385067-1

Integration of functions with values in locally convex Suslin spaces
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by G. Erik F. Thomas

Trans. Amer. Math. Soc. 212 (1975), 61-81

DOI: https://doi.org/10.1090/S0002-9947-1975-0385067-1

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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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