Measurable selectors for the metric projection
by
B. Cascales
Universidad de Murcia
Coauthors: M. Raja (Universidad de Murcia)

A closed subspace Y of a Banach space (X, || ||) is said to be proximinal if for every x in X there is y in Y such that ||x-y||=d(x, Y):=inf{||x-z||: z in Y}; if Y is proximinal the set P_Y(x):={y in Y: ||x-y||=d(x, Y)} is closed and convex for every x in X. It is easy to check that if we do assume Y being reflexive then the metric projection P_Y: X --> 2^Y is upper semicontinuous weakly compact valued and therefore has a first Baire class selector f, that is, there is a single valued map f: X --> Y that is the pointwise limit of a sequence of norm to norm continuous functions (f is first Baire class) and f(x) belongs to P_Y(x) for every x in X (f is a selector for P_Y). We shall show that when Y is weakly countably determined and proximinal (in particular reflexive or separable and proximinal or WCG and proximinal, etc.) the metric projection P_Y restricted to separable subspaces M of X does have a selector that is measurable with respect to the \sigma-algebra generated in M by the analytic sets; by doing so we obtain that for Y weakly countably determined proximinal, for every complete probability space (\Omega, \Sigma, \mu) and every p between 1 and \infty the space L^p(\mu, Y) is proximinal in (L^p(\mu, X), ||  ||_p) which unifies and non trivially extends results dating from 1981 (by Light and Cheney) until 1998 (by Mendoza). These results here are even proved in a more general setting of metric spaces and functions in two variables that can be also applied to describe a certain structure of Bishop-Phelps sets and to re-obtain a characterization of RNP in dual Banach spaces via measurable selectors for the duality map.

(T)

Date received: April 14, 2000