Integration of Functions With Values in Locally Convex Suslin Spaces

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 ⊂ E’ is any total subset, and f is an E-valued function which is Pettis summable relative to the ultra weak topology σ (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.