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The weak McShane integral. (English) Zbl 1340.28016
The authors introduce a modification of the generalized McShane integration process due to D. H. Fremlin [Ill. J. Math. 39, No. 1, 39–67 (1995; Zbl 0810.28006)] that involves Banach space-valued integrands on the \(\sigma \)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal {T},\mu)\). This new integral, the weak McShane integral in the author’s terminology, is defined in the same manner as the Fremlin integral to be the gauge limit of the generalized McShane sums in the weak topology, rather than in the norm topology.
The key results of the paper under review compare the weak McShane integral with the Pettis integral:
A function is weakly McShane integrable on each measurable subset of \(S\) if and only if it is both Pettis and weakly McShane integrable on \(S\).
If a function is weakly McShane integrable on \(S\), then there exist measurable sets of finite measure \((S_{l})_{l\geq 1}\) that cover \(S\) on each of which the function is Pettis integrable.
If the range does not contain an isomorphic copy of \(c_{0}\), then each weakly McShane integrable function is necessarily Pettis integrable. Moreover, the assumption that the range does not contain an isomorphic copy of \(c_{0}\) cannot be removed from this theorem.

MSC:
28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
26A39 Denjoy and Perron integrals, other special integrals
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