## 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.)

### Citations:

Zbl 0482.28011; Zbl 0516.28011
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