Beppo Levi’s theorem for the vector-valued McShane integral and applications. (English) Zbl 1038.46505

Swartz, Charles

Bull. Belg. Math. Soc. - Simon Stevin 4, No. 5, 589-599 (1997).

Summary: Using only elementary properties of the McShane integral for vector-valued functions, we establish a convergence theorem which for the scalar case of the integral yields the classical Beppo Levi (monotone) convergence theorem as an immediate corollary. As an application, the convergence theorem is used to prove that the space of McShane integrable functions, although not usually complete, is ultrabornological.

MSC:

46E40

Spaces of vector- and operator-valued functions

Keywords:

barrelled; ultrabornological