(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!