Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-13T08:00:39.107Z Has data issue: false hasContentIssue false
  • English
  • Français

Spaces of vector functions that are integrable with respect to vector measures

Part of: Topological linear spaces and related structures Linear function spaces and their duals Measures, integration, derivative, holomorphy Set functions, measures and integrals with values in abstract spaces

Published online by Cambridge University Press:  09 April 2009

José Rodríguez
José Rodríguez
Affiliation:
Departamento de Matemáticas Universidad de Murcia30.100 Espinardo MurciaSpain e-mail: joserr@um.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the normed spaces of (equivalence classes of) Banach space-valued functions that are Dobrakov, S* or McShane integrable with respect to a Banach space-valued measure, where the norm is the natural one given by the total semivariation of the indefinite integral. We show that simple functions are dense in these spaces. As a consequence we characterize when the corresponding indefinite integrals have norm relatively compact range. On the other hand, we also determine when these spaces are ultrabornological. Our results apply to conclude, for instance, that the spaces of Birkhoff (respectively McShane) integrable functions defined on a complete (respectively quasi-Radon) probability space, endowed with the Pettis norm, are ultrabornological.

Keywords

Dobrakov integral; S*-integral; McShane integral; Birkhoff integral; ultrabornological space; barrelled space.

MSC classification

Secondary: 28B05: Vector-valued set functions, measures and integrals 46G10: Vector-valued measures and integration 46A08: Barrelled spaces, bornological spaces 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 1 , February 2007 , pp. 85 - 109
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bartle, R. G., ‘A general bilinear vector integral’, Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
[2]Birkhoff, G., ‘Integration of functions with values in a Banach space’, Trans. Amer. Math. Soc. 38 (1935), 357–378.Google Scholar
[3]Cascales, B. and Rodriguez, J., ‘The Birkhoff integral and the property of Bourgain’, Math. Ann. 331 (2005), 259–279.CrossRefGoogle Scholar
[4]Díaz, S., Fernández, A., Florencio, M. and Paúl, P. J., ‘A wide class of ultrabornological spaces of measurable functions’, J. Math. Anal. Appl., 190 (1995), 697–713.CrossRefGoogle Scholar
[5]Diestel, J. and Uhl, J. J. Jr, Vector measures, Math. Surveys 15 (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
[6]Dobrakov, I., ‘On integration in Banach spaces. I’, Czechoslovak Math. J. 20 (95) (1970), 511–536.CrossRefGoogle Scholar
[7]Dobrakov, I., ‘On representation of linear operators on C 0(T, X)’, Czechoslovak Math. J. 21 (96) (1971), 13–30.CrossRefGoogle Scholar
[8]Dobrakov, I., ‘On integration in Banach spaces. VII’, Czechoslovak Math. J. 38 (113) (1988), 434–449.CrossRefGoogle Scholar
[9]Fremlin, D. H., ‘The McShane and Birkhoff integrals of vector-valued functions’, Research report 92–10, version of 13.10.04(Mathematics Department, University of Essex, Colchester, UK), available at http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm.Google Scholar
[10]Fremlin, D. H., ‘Problem ET’, version of 27.10.04 (Mathematics Department, University of Essex, Colchester, UK), available at http://www.essex.ac.uk/maths/staff/fremlin/problems.htm.Google Scholar
[11]Fremlin, D. H., ‘The generalized McShane integral’, Illinois J. Math. 39 (1995), 39–67.CrossRefGoogle Scholar
[12]Fremlin, D. H., Measure theory. Vol. 4 Topological measure spaces (Torres Fremlin, Colchester, 2003).Google Scholar
[13]Fremlin, D. H. and Talagrand, M., ‘A decomposition theorem for additive set-functions, with applications to Pettis integrals and ergodic means’, Math. Z. 168 (1979), 117–142.CrossRefGoogle Scholar
[14]Freniche, F. J. and García-Vázquez, J. C., ‘The Bartle bilinear integration and Carleman operators’, J. Math. Anal. Appl. 240 (1999), 324–339.CrossRefGoogle Scholar
[15]Gámez, J. L., Denjoy integrals of functions with values in Banach spaces (Ph.D. Thesis, Universidad Complutense de Madrid, Spain, 1997), (Spanish).Google Scholar
[16]Gilioli, A., ‘Natural ultrabornological, noncomplete, normed function spaces’, Arch. Math. (Basel) 61 (1993), 465–477, Edited by Klaus Floret and Chaim S. Hönig.CrossRefGoogle Scholar
[17]Janicka, L. and Kalton, N. J., ‘Vector measures of infinite variation’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 239–241.Google Scholar
[18]Jarchow, H., Locally convex spaces (B. G. Teubner, Stuttgart, 1981).CrossRefGoogle Scholar
[19]Musial, K., ‘Topics in the theory of Pettis integration’, Rend. Istit. Mat. Univ. Trieste 23 (1991), 177–262.Google Scholar
[20]Panchapagesan, T. V., ‘On the distinguishing features of the Dobrakov integral’, Divulg. Mat. 3 (1995), 79–114.Google Scholar
[21]Paúl, P. J., ‘The space of Denjoy-Dunford integrable functions is ultrabornological’, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 75–82.Google Scholar
[22]Carreras, P. Pérez and Bonet, J., Barrelled locally convex spaces, North-Holland Math. Studies 131 (North-Holland, Amsterdam, 1987).Google Scholar
[23]Pettis, B. J., ‘On integration in vector spaces’, Trans. Amer. Math. Soc. 44 (1938), 277–304.CrossRefGoogle Scholar
[24]Di Piazza, L. and Presis, D., ‘When do McShane and Pettis integrals coincide?’, Illinois J. Math. 47 (2003), 1177–1187.CrossRefGoogle Scholar
[25]Rodríguez, J., ‘On the existence of Pettis integrable functions which are not Birkhoff integrable’, Proc. Amer. Math. Soc. 133 (2005), 1157–1163.CrossRefGoogle Scholar
[26]Rodríguez, J., ‘On integration of vector functions with respect to vector measures’, Czechoslovak Math. J. 56 (2006), 805–825.CrossRefGoogle Scholar
[27]Swartz, C., ‘Beppo Levi's theorem for the vector-valued McShane integral and applications’, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 589–599.Google Scholar
[28]Swartz, C., ‘Uniform integrability and mean convergence for the vector-valued McShane integral’, Real Anal. Exchange 23 (1997/1998), 303–311.CrossRefGoogle Scholar
[29]Swartz, C., ‘Barrelledness of the space of Dobrakov integrable functions’, Math. Slovaca 51 (2001), 521–528.Google Scholar
[30]Thomas, G. E. F., ‘Totally summable functions with values in locally convex spaces’, in: Measure theory (Proc. Conf., Oberwolfach, 1975), Lecture Notes in Math. 541 (Springer, Berlin, 1976) pp.117–131.Google Scholar