Absolutely summing operators and integration of vector-valued functions

Let (Ω,Σ,μ) be a complete probability space and u:X →Y an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function f :Ω →X the composition u ◦ f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f →u ◦ f . Some other applications are given.  2005 Elsevier Inc. All rights reserved.