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Boundedness of vector measures with values in the spaces \(L_ 0\) of Bochner measurable functions. (English) Zbl 0601.28007

Let \(L_ 0(Z)\) be the F-space of all Bochner measurable functions from a probability space to a Banach space Z. We prove that every countably additive vector measure taking values in \(L_ 0(Z)\) has bounded range. This generalizes a recent result due to M. Talagrand [Ann. Sci. Ec. Norm. Supér., IV. Sér. 14, 445-452 (1981; Zbl 0482.28011)] and, independently, N. J. Kalton, N. T. Peck and J. W. Roberts [Proc. Am. Math. Soc. 85, 575-582 (1982; Zbl 0516.28011)] asserting the same for the case when Z is the space of scalars.

MSC:

28B05 Vector-valued set functions, measures and integrals
46G10 Vector-valued measures and integration
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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