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When \(L^ 1\) of a vector measure is an AL-space. (English) Zbl 0791.46021

We consider the space of real functions which are integrable with respect to a countably additive vector measure with values in a Banach space, in the sense of R. G. Bartle, N. Dunford and J. T. Schwartz [Can. J. Math. 7, 289-305 (1955; Zbl 0068.093)]. In a previous paper [Math. Ann. 293, No. 2, 317-330 (1992; Zbl 0782.46042)], we show that this space can be any order continuous Banach lattice with weak order unit. We study a priori conditions on the vector measure in order to guarantee that the resulting \(L^ 1\) is order isomorphic to an AL- space. We prove that for separable measures with no atoms there exists a \(c_ 0\)-valued measure that generates the same space of integrable functions.

MSC:

46G10 Vector-valued measures and integration
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B40 Ordered normed spaces
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