Publication: On the integration of vector-valued functions.
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1994-03
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University of Illinois
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Abstract
As the authors of this article state, "the ordinary functional analyst is naturally impatient with the multiplicity of definitions of `integral' which have been proposed for vector-valued functions, and would much prefer to have a single canonical one for general use''. Such a goal is unlikely to be attained, however. While the Bochner integral has been extensively studied and exploited in applications, other types of integrals have had to be introduced to extend the class of integrable functions. In this paper, the authors study three such extensions: the Pettis integral, the Talagrand integral, and the McShane integral. The first of these is well known, the latter two perhaps not as a familiar.
A function φ:S→X from a probability space to a Banach space is Talagrand integrable, with Talagrand integral w, if w=limn→∞(1/n)∑i<nφ(si) for almost all sequences (si) in SN, where SN is given its product probability.
A function φ:[0,1]→X is McShane integrable, with McShane integral w, if for every ε>0 there is a gauge function δ:[0,1]→(0,∞) such that ∥w−∑i≤n(bi−ai)φ(ti)∥≤ε for every finite sequence ([ai,bi],ti)i≤n of a nonoverlapping family of intervals covering [0,1] and ti∈[0,1] satisfying ti−δ(ti)≤ai≤bi≤ti+δ(ti).
The paper investigates the relationships between these three integrals and the Bochner integral. These relationships depend on conditions imposed on both the Banach space and the function φ. Some of the new results contained here are that, with no restrictions on the space or the function, a McShane integrable function is Pettis integrable. If the unit ball of the dual space of X is separable, then a McShane integrable function must be Talagrand integrable. If X is separable but we allow the possibility that φ is unbounded, then the Bochner and Talagrand integrals coincide and the McShane and Pettis integrals coincide, while if X is separable and φ is bounded, then all four integrals coincide. Examples are given of a bounded McShane integrable function from [0,1] to l∞(c) and an unbounded McShane integrable function from [0,1] to l2(N) that are not Talagrand integrable, and of a bounded Talagrand integrable function from [0,1] to l∞(N) that is not McShane integrable.
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