SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £∞ ((O, 1), X) → £∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X. As an application we introduce the space £∞ strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £∞ strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £∞(μ Y**)). Finally we investigate the measurability property of functions in £∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).