Abstract

We construct, over many Banach spaces, infinite dimensional vector spaces of scalarly measurable functions that are not strongly measurable, and infinite dimensional vector spaces of ω∗\omega^*-scalarly measurable functions that are not scalarly measurable. A similar result will be proved for the set of McShane-integrable functions which are not Bochner-integrable.

Author information

Contact details are reproduced from the original publication and may be historical.

Francisco J. García-Pacheco

Dept. of Mathematics, University of Cadiz, 11510 Puerto Real, Spain

garcia.pacheco@uca.es

Keywords

  • Strongly measurable
  • scalarly measurable
  • omega-star-scalarly measurable
  • McShane integrable functions
  • Bochner integrable functions

References

  1. J. Diestel, J. J. Uhl jr.: Vector Measures, Mathematical Surveys and Monographs 15, AMS, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  2. J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators, Cambridge University Press, Cambridge (1995).https://doi.org/10.1017/CBO9780511526138Google Scholar
  3. M. Fabian et al.: Functional Analysis and Infinite-Dimensional Geometry, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3480-5Google Scholar
  4. D. H. Fremlin, M. Talagrand: A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means, Math. Z. 168(2) (1979) 117–142.https://doi.org/10.1007/BF01214191Google Scholar
  5. R. A. Gordon: The McShane integral of Banach-valued functions, Ill. J. Math. 34 (1990) 557–567.https://doi.org/10.1215/ijm/1255988170Google Scholar

Suggested citation

F. J. García-Pacheco, M. A. Sofi. “Lineability Issues Involving Vector-Valued Measurable and Integrable Functions.” Journal of Convex Analysis 19 (2012), No. 2, 393–402. https://doi.org/10.68381/jca19022

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.