Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. If all X-valued Pettis integrals defined on (Ω,Σ,μ) have separable ranges we show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of c0 if and only if X does.
[1] Bourgain, J. 1978 An averaging result for c0-sequences Bull. Soc. Math. Belg. 30 83–87.
[2] Cembranos, P. Mendoza, J. 1997 Banach Spaces of Vector-Valued Functions Lecture Notes in Math. 1676 Springer.
[3] Diestel, J. Uhl, J. 1977 Vector Measures Math. Surveys 15 Amer. Math. Soc. Providence.
[4] Drewnowski, L. 1990 When does ca(Σ,Y) contain a copy of ℓ∞ or c0? Proc. Amer. Math. Soc. 109 747–752.
[5] Drewnowski, L. Florencio, M. Paúl, P. J. 1992 The space of Pettis integrable functions is barrelled Proc. Amer. Math. Soc. 114 687–694 .
[6] Dunford, N. Schwartz, J. T. 1988 Linear Operators, Part I. General Theory John Wiley & Sons New York, Chichester, Brisbane, Toronto, Singapore.
[7] Ferrando, J. C. 1994 When does bvca (Σ,X) contain a copy of ℓ∞? Math. Scand. 74 271–274.
[8] Ferrando, J. C. 2002 On sums of Pettis integrable random elements Quaestiones Math. 25 311–316 .
[9] Ferrando, J. C. Sánchez Ruiz, L. M. 2007 Embedding c0 in bvca (Σ,X) Czech. Math. J. 57 679–688 .
[10] Freniche, F. J. 1998 Embedding c0 in the space of Pettis integrable functions Quaestiones Math. 21 261–267 .
[11] Kwapień, S. 1974 On Banach spaces containing c0 Studia Math. 52 187–188.
[12] Musial, K., Pettis integration, in: Proc. 13th Winter School on Abstract Analysis, Supplemento Rend. Circolo Mat. Palermo, Serie II, (10) (1985), pp. 133–142.
[13] Musial, K. 2002 Pettis integral Handbook of Measure Theory 531–586 .
[14] Saab, E. Saab, P. 1986 On complemented copies of c0 in injective tensor products Contemp. Math. 52 131–135 .
[15] Talagrand, M. 1980 Sur les mesures vectorielles définies par une application Pettis intégrable Bull. Soc. Math. France 108 475–483.
[16] Talagrand, M. 1984 Quand l'espace des mesures à variation bornée est-il faiblement séquentiellement complet? Proc. Amer. Math. Soc. 90 285–288.