# Integration with values in a topological vector space

Is there a general theory of integration of functions with values in a topological vector space (not necessarily locally convex)?

Browsing through mathoverflow posts, I came across a discussion regarding Grothendieck's work on integration with values in a topological group L'intégration à valeurs dans un groupe topologique (Reference request : Grothendieck's topological space valued integral ). Unfortunately the requested reference is never provided. Does anyone know of a rough sketch of the approach used?

Just to emphasize, I am interested in a general theory and not in notions such as the Bochner integral, or Gelfand-Pettis integral which use local convexity of the topological vector space.

• A hint of the kind of negative things that can happen: it is routine to extend the Riemann integral to this situation but it can happen that a continous function, say on $[0,1]$, is not integrable. The basic reason is that in a non locally convex space, convex combinations of small elements can be huge, and Riemann sums are just such combinations. – user131781 Jul 10 at 18:26
• However, there was some work done on this topic, in particular for so-called locally bounded tvs‘s, about 50 years ago. I suggest you look up D. Voigt, B. Gramsch and L. Waelbroeck in suitable sources (MathSciNet, Zentralblatt, etc.). – user131781 Jul 10 at 18:37
• @user131781: That's a good point. A reference I found numdam.org/article/PSMIR_1979___1_A3_0.pdf (Butković, D. On integration with respect to measures with values in arbitrary topological vector spaces. Glas. Mat. Ser. III 15(35) (1980), no. 1, 33–40. ) This is along the lines of Gerald Edgar's answer. This could get me started in understanding these measures. But I am still looking for a direct answer to integrating vector-valued functions over real or complex measures rather than vector-valued measures. – TVS_integration Jul 11 at 19:20
• That is exactly what is done by the authors I mentioned. – user131781 Jul 11 at 20:02

## 1 Answer

One reference ...

Sion, Maurice, A theory of semigroup valued measures, Lecture Notes in Mathematics. 355. Berlin-Heidelberg-New York: Springer-Verlag. V, 140 p. DM 16.00; $6.60 (1973). ZBL0312.28016. • Thank you. I edited the question a bit to emphasize that I want the function to take values in a topological vector space (and the measure is real or complex-valued). – TVS_integration Jul 11 at 14:05 • If$\mu$is a real-valued measure, and we are doing an integral like$\int_A f\;d\mu$where$f$has values in a tvs$X$, then the set-function$A \mapsto \int_A f\;d\mu$is a measure with values in$X\$. Thus, the theory of vector-valued integrals is related to the theory of vector-valued measures. – Gerald Edgar Jul 11 at 14:09
• You are right. They are related. A theory of vector-valued integrals immediately leads to a theory of vector-valued measure. But it is hard for me to visualize the opposite direction, that is, if we have a theory of vector-valued measures, how do we recover a theory of integration of vector-valued functions? Is there a recipe for doing that? (It invokes a feeling of some type of Radon-Nikodym there but I can't quite put my finger to it.) – TVS_integration Jul 11 at 14:47
• @TVS_integration The Radon-Nikodym theorem can fail already for Banach spaces, so there's no need to go as far as non-locally convex spaces for that. The Banach spaces for which it doesn't fail are said to have "the Radon-Nikodym property" and it can be related to all sorts of other things. – Robert Furber Jul 11 at 15:51