Functiones et Approximatio Commentarii Mathematici

On functions that are $BDS$-integrable over convexly bounded vector measures

Iwo Labuda

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Abstract

Spaces of scalar functions that are integrable in the sense of Bartle-Dunford-Schwartz integration, with respect to a~convexly bounded vector measure $\mu$, are studied. For instance, under the assumption that the range space $X$ of $\mu$ is sequentially complete, the effect of the Orlicz-Pettis property (with respect to a weaker topology on $X$) on the size of $L^1(\mu)$ is investigated. Some completeness properties of the space $L^1_\bullet(\mu)$ of `scalarly integrable functions' are established for general $X$.

Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 1 (2014), 151-159.

Dates
First available in Project Euclid: 27 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1395924288

Digital Object Identifier
doi:10.7169/facm/2014.50.1.4

Mathematical Reviews number (MathSciNet)
MR3189504

Zentralblatt MATH identifier
1292.28018

Subjects
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]
Secondary: 46A40: Ordered topological linear spaces, vector lattices [See also 06F20, 46B40, 46B42]

Keywords
convexly bounded vector measure Bartle-Dunford-Schwartz integration spaces of integrable functions $\sigma$-Lebesgue property $\sigma$-Levi property

Citation

Labuda, Iwo. On functions that are $BDS$-integrable over convexly bounded vector measures. Funct. Approx. Comment. Math. 50 (2014), no. 1, 151--159. doi:10.7169/facm/2014.50.1.4. https://projecteuclid.org/euclid.facm/1395924288


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References

  • C. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, 2nd ed., Math. Surveys and Monographs, vol. 105, Amer. Math. Soc., 2003.
  • L. Drewnowski and I. Labuda, Bartle-Dunford-Schwartz Integration, J. Math. Anal. Appl. 401 (2013), 620–640.
  • S. Rolewicz, Metric Linear Spaces, Polish Scientific Publishers, Warszawa, 1972.
  • S. Rolewicz, Metric Linear Spaces, Polish Scientific Publishers & D. Reidel Publishing Co., Warszawa & Dordrecht, Boston, Lancaster, 1984.
  • E. Thomas, L',intégration par rapport à une mesure de Radon vectorielle, Ann. Inst. Fourier 20 (1970), 55–191.
  • E. Thomas, On Radon maps with values in arbitrary topological vector spaces, and their integral extensions, Department of Mathematics, Yale University preprint (Unpublished), pages 1–78, 1972.
  • Ph. Turpin, Intégration par rapport à une mesure à valeurs dans un espace vectoriel topologiques non supposé localement convexe, in: Actes de Colloque `Intégration vectorielle et multivoque', Université de Caen, 22 et 23 mai 1975, pp. 8.1–8.21.
  • Ph. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976), 1–221.