On some subsets of L 1 (μ,E)

Abstract

The paper starts with the following remark: One of the most common methods used in the literature to introduce new properties in a Banach space E consists in establishing some nontrivial relationships between different classes of subsets of E . Moving on from this, the author considers the classes of bounded, weakly relatively compact, weakly conditionally compact, norm relatively compact, Dunford-Pettis, and (V* ) subsets of L 1 (μ,E) (in symbols: B,W,WC,K,DP,V* , respectively) and investigates their nature and the consequences of the possible coincidence of two of them in terms of properties of the space L 1 (μ,E) . He observes that the following necessary condition is true. Proposition II.1: Let H be any of the classes K,W,WC,DP and V* . If M H(L 1 (μ,E)) then: (a) M is bounded; (b) M is uniformly integrable; (c) for every measurable set A , M(A)={∫ A fdμ , f K} is in H(E) . Then he gives the following definition: A subset M of L 1 (μ,E) satisfying conditions (a) to (c) of Proposition II.1 is called a μH -set; a Banach space E is said to have property P(μ,H) if every μH -set belongs to H(L 1 (μ,E)) . Then he gives necessary and sufficient conditions for a Banach space E to have property P(μ,V*),P(μ,WC) and P(μ,DP)