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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Radon-Nikodým derivatives for Banach lattice-valued measures


Author: Ep de Jonge
Journal: Proc. Amer. Math. Soc. 83 (1981), 489-495
MSC: Primary 28B05; Secondary 46G10
MathSciNet review: 627676
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Abstract: Let $ (\Delta ,\Gamma ,\mu )$ be a measure space such that $ 0 < \mu (\Delta ) < \infty $ and such that $ \Gamma $ has no $ \mu $-atoms. Furthermore, let $ E$ be a Dedekind complete Banach lattice. By $ M(\mu ,E)$ we denote the set of all $ E$-valued set functions $ \nu $ on $ \Gamma $ satisfying

(i) $ \nu $ is additive,

(ii) $ \nu $ is order bounded and of bounded variation,

(iii) $ \nu $ is $ \mu $-absolutely continuous (with respect to the norm topology on $ E$),

(iv) $ {\nu ^ + }$ and $ {\nu ^ - }$ satisfy

$\displaystyle \inf \{ \sup \{ {\upsilon ^{ + / - }}(A):\mu (A) < \varepsilon \} :\varepsilon > 0\} = 0.$

By $ {L_1}(\mu ,E)$ we denote the set of $ E$-valued Bochner integrable functions on $ \Delta $. It is shown that under the canonical map

$\displaystyle f \mapsto {\nu _f}$

(where $ {\nu _f}(A) = \smallint f{\mathcal{X}_A}d\mu $, $ A \in \Gamma $), $ {L_1}(\mu ,E)$ is a Riesz subspace of the Dedekind complete Riesz space $ M(\mu ,E)$. Furthermore, the following theorems are proved.

Theorem. Equivalent are

(a) $ {l_\infty }$ is not isomorphic to a closed sublattice of $ E$.

(b) $ {L_1}(\mu ,E)$ is isomorphic to an ideal in $ M(\mu ,E)$.

Theorem. Equivalent are

(a) $ {c_0}$ is not isomorphic to a closed sublattice of $ E$.

(b) $ {L_1}(\mu ,E)$ is isomorphic to $ M(\mu ,E)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0627676-6
PII: S 0002-9939(1981)0627676-6
Keywords: Radon-Nikodym derivatives, Bochner integral, Banach lattices
Article copyright: © Copyright 1981 American Mathematical Society



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