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Darboux property of the nonatomic σ-additive positive and finite dimensional vector measures

Lorenc P., Wituła R.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

2013

z. 3

25--36

In the paper some facts connected with Darboux property of the positive measures and the finite dimensional vector measures are discussed.

W artykule omawiane są pewne fakty związane z własnością Darboux miar nieujemnych i miar wektorowych skończenie wymiarowych.

Vector measures, c0, and (sb) operators

Bator E. M., Lewis P. W., Slavens D. R.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 1

63-73

Emmanuele showed that if Σ is a σ-algebra of sets, Χ is a Banach space, and μ : Σ → Χ is countably additive with finite variation, then μ(Σ) is a Dunford-Pettis set. An extension of this theorem to the setting of bounded and finitely additive vector measures is established. A new characterization of strongly bounded operators on abstract continuous function spaces is given. This characterization motivates the study of the set of (sb) operators. This class of maps is used to extend results of P. Saab dealing with unconditionally converging operators. A characterization of the existence of a countably additive, non-strongly bounded representing measure in terms of c0 is presented. This characterization resolves a question posed in 1970.