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(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", "Gelfand-Pettis integral", and "weak integral", and so far I have been unable to find something entirely satisfactory. I would like to find a more comprehensive source than those I've been finding.

In particular, I am interested in finding suitable existence theorems. The one's I've been finding (e.g. Garrett's Notes) seem to have pretty strong hypotheses. Specifically, my integration space is not of finite measure nor or the functions I am interested in integrating compactly supported. (I am thinking of functions on the real line that take values in the space of Schwartz functions and the space of tempered distributions.)

Do you know of any sources that have results along these lines?

Thanks in advance!

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4 Answers 4

There are some results of the kind you look for, in Section 8.14 of the book

R.E. Edwards, Functional Analysis, Holt, Rinehart and Winston, New York, 1965.

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It sounds as if you are after the theory of the Bochner integral. The classical text of Diestel and Uhl ("Vector measures") has the basic material on this. It may be that they only consider finite measures but, even if so, the extension to the case of $ \sigma$-finite measures or (unbounded) Radon measures on a locally compact space is routine. They are also mostly interested in Banach spaces but the generalisation to complete locally convex spaces, especially if separable as in your case, is also rather simple. In fact, if you are interested in finer properties (I am thinking of Radon-Nikodym type results), then the fact that your spaces are Fréchet nuclear spaces or nuclear Silva spaces means that your situation will be simpler and the results stronger. If, as your title (but not the text of your question) suggests, you are interested in the case of topological linear spaces, then be warned that this is a different kettle of fish. In the non locally convex case, it can happen that continuous functions, say on the unit interval, are not integrable with respect to Lebesgue measure. The problem lies in the fact that in this situation, convex combinations of small vectors can be very large. You could consult the works of Bernhard Gramsch and collaborators for this.

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Some useful cases of non-compactly-supported, but continuous $f$, on not-finite-measure spaces, can be reduced to the compactly-supported, finite-measure case by a compactification and renormalization. For example, in an integral $\int_{\mathbb R} f(t)\,dt$ with $V$-valued $f$, for $\varepsilon>0$ rewrite this as $\int_{\mathbb R} (1+t^2)^{{1\over 2}+\varepsilon}\,f(t)\cdot {dt\over (1+t^2){^{1\over 2}+\varepsilon}}$. First, with this new measure, the real line (and/or compactified by adding a point at infinity) has finite mass. Then, if $(1+t^2)^{{1\over 2}+\varepsilon}\cdot f(t)$ extends to (e.g.) the one-point compactification, as a continuous $V$-valued function, then the finite-measure, compact-support hypotheses for Gelfand-Pettis integrals hold, and we reach the same conclusion.

(In fact, Schwartz' original discussion of what we call the Schwartz space overtly used a certain compactification of $\mathbb R^n$ to an $n$-sphere, and Schwartz functions were those which extended differentiably and vanished to infinite order at the point at infinity.)

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survey as of 1940 ...

R. S. Phillips, Integration in a convex linear topological space, Trans. Amer. Math. Soc. 47 (1940) 114--145

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