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On the Banach lattice structure of \(L^1_w\) of a vector measure on a \(\delta \)-ring

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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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References

  1. Brooks, J.K., Dinculeanu, N.: Strong additivity, absolute continuity and compactness in spaces of measures. J. Math. Anal. Appl. 45, 156–175 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  2. Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Calabuig, J.M., Juan, M.A., Sánchez Pérez, E.A.: Spaces of \(p\)-integrable functions with respect to a vector measure defined on a \(\delta \)-ring. Oper. Matrices 6, 241–262 (2012)

  4. Curbera, G.P.: El espacio de funciones integrables respecto de una medida vectorial. Ph. D. thesis, University of Sevilla, Sevilla (1992)

  5. Curbera, G.P.: Operators into \(L^1\) of a vector measure and applications to Banach lattices. Math. Ann. 293, 317–330 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Curbera, G.P., Ricker, W.J.: Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17, 187–204 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. G. P. Curbera and W. J. Ricker, Vector measures, integration and applications. In: Positivity (in Trends Math.), Birkhäuser, Basel, pp. 127–160 (2007)

  8. Curbera, G.P., Ricker, W.J.: The Fatou property in \(p\)-convex Banach lattices. J. Math. Anal. Appl. 328, 287–294 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Delgado, O.: \(L^1\)-spaces of vector measures defined on \(\delta \)-rings. Arch. Math. 84, 432–443 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Delgado, O.: Optimal domains for kernel operators on \([0,\infty )\times [0,\infty )\). Studia Math. 174, 131–145 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Delgado, O., Juan, M.A.: Representation of Banach lattices as \(L_w^1\) spaces of a vector measure defined on a \(\delta \)-ring. Bull. Belg. Math. Soc. Simon Stevin 19(2), 239–256 (2012)

    MATH  MathSciNet  Google Scholar 

  13. Diestel, J., Uhl, J.J.: Vector measures (Am. Math. Soc. surveys 15). American Mathematical Society, Providence (1997)

    Google Scholar 

  14. Dinculeanu, N.: Vector measures, Hochschulbcher fr Mathematik, vol. 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)

    Google Scholar 

  15. Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., Sánchez Pérez, E.A.: Spaces of \(p\)-integrable functions with respect to a vector measure. Positivity 10, 1–16 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fremlin, D.H.: Measure theory, broad foundations, vol. 2. Torres Fremlin, Colchester (2001)

    Google Scholar 

  17. Jiménez Fernández, E., Juan, M.A., Sánchez Pérez, E.A.: A Komlós theorem for abstract Banach lattices of measurable functions. J. Math. Anal. Appl. 383, 130–136 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lewis, D.R.: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)

    MATH  Google Scholar 

  19. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  20. Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  21. Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on \(\delta \)-rings. Adv. Math. 73, 204–241 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Masani, P.R., Niemi, H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121–167 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Thomas, E.G.F.: Vector integration (unpublished) (2013)

  24. Turpin, Ph.: Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologique non supposé localement convexe, Intègration vectorielle et multivoque, (Colloq., University Caen, Caen, 1975), experiment no. 8, Dèp. Math., UER Sci., University Caen, Caen (1975)

  25. Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces (Oper. Theory Adv. Appl.), vol. 180. Birkhäuser, Basel (2008)

    Google Scholar 

  26. Zaanen, A.C.: Riesz spaces II. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022, Valencia, Spain

    J. M. Calabuig, M. A. Juan & E. A. Sánchez Pérez

  2. Departamento de Matemática Aplicada I, E. T. S. de Ingeniería de Edificación, Universidad de Sevilla, Avenida Reina Mercedes, 4 A, 41012, Sevilla, Spain

    O. Delgado

Authors
  1. J. M. Calabuig
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  2. O. Delgado
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  3. M. A. Juan
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  4. E. A. Sánchez Pérez
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Correspondence to J. M. Calabuig.

Additional information

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

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