Boundedness of vector measures with values in the spaces 𝐿₀ of Bochner measurable functions

Drewnowski, Lech

doi:10.1090/S0002-9939-1984-0746094-6

Boundedness of vector measures with values in the spaces $L_{0}$ of Bochner measurable functions
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Proc. Amer. Math. Soc. 91 (1984), 581-588 Request permission

Abstract:

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 and, independently, N. J. Kalton, N. T. Peck and J. W. Roberts, asserting the same for the case when $Z$ is the space of scalars.

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