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On the equivalence of McShane and Pettis integrability in non-separable Banach spaces. (English) Zbl 1138.28003
In this paper the author considers the relationship between the McShane, Pettis and Bochner integrals for functions with values in a Banach space. The McShane integral lies properly between the Pettis and Bochner integrals and in certain spaces such as separable Banach spaces or Hilbert spaces the McShane and Pettis integrals coincide. The author shows that for functions \( f:[0,1]\rightarrow L^{1}(\mu )\) , \(\mu \) any finite measure, the McShane and Pettis integrals coincide. Assuming the Continuum Hypothesis, the author gives an example of a weakly Lindelöf determined Banach space \(X\) , a function \(f:[0,1]\rightarrow X\) and an absolutely summing operator \( u:X\rightarrow Y\), a Banach space, such that \(f\) is scalarly null (so Pettis integrable) and \(uf\) is not Bochner integrable so \(f\) is not McShane integrable.

MSC:
28B05 Vector-valued set functions, measures and integrals
46G15 Functional analytic lifting theory
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
28B15 Set functions, measures and integrals with values in ordered spaces
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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