Abstract
We study some Banach lattice properties of the space \(L_w^1(\nu )\) of weakly integrable functions with respect to a vector measure \(\nu \) defined on a \(\delta \)-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of \(L_w^1(\nu )\) differs from the case in which \(\nu \) is defined on a \(\sigma \)-algebra whenever \(\nu \) does not satisfy certain local \(\sigma \)-finiteness property.
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J. M. Calabuig and M. A. Juan were supported by the Ministerio de Economía y Competitividad (project MTM2008-04594). O. Delgado was supported by the Ministerio de Economía y Competitividad (project MTM2009-12740-C03-02). E. A. Sánchez Pérez was supported by the Ministerio de Economía y Competitividad (project MTM2009-14483-C02-02).
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Calabuig, J.M., Delgado, O., Juan, M.A. et al. On the Banach lattice structure of \(L^1_w\) of a vector measure on a \(\delta \)-ring. Collect. Math. 65, 67–85 (2014). https://doi.org/10.1007/s13348-013-0081-8
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DOI: https://doi.org/10.1007/s13348-013-0081-8
Keywords
- Banach lattice
- \(\delta \)-ring
- Fatou property
- Order density
- Order continuity
- Integration with respect to vector measures