ON CARATH EO D O RY TYPE SELECTORS

Abstract. In this paper we consider a set-valued function of two variables, measurable in the first and continuous in the second variable. Using metric projections we construct for this function a family of selectors which are Caratheodory maps. The existence of Caratheodory selectors was studied by Castaing [2], [3], Cellina [4], Fryszkowski [9] and the first author [11]. 1. Notation and definitions. Let (T, ST) be a measurable space, X a topological space and Y a Hilbert space. By 0>C(Y) we denote the family of all non-empty closed convex subsets of Y. We shall consider 9 C ( Y) with the Vietoris topology (see e.g. [12, § 17.1]), and with the generalized Hausdorff metric d is t ( / l, B) = sup { d ( a, B) , d ( b, A) : a e A, b e B}, where d(a, B) = inf {|| « — ft| | : b e B}, A, B e.? c(Y) (we admit dist(/4, B) = oo). Let y 0 be a point of Y and r a positive number. By B (y0, r) (E(y0, /•)) we denote the open (closed) ball with centre y 0 and radius r. For a set A c Y and r> 0, B(A, r) denotes the /--ball about A. Let ę: T ~ * &C{Y) be a multifunction (i.e. set-valued mapping). A function f: T-> Y is a selector for cp if / ( / ) e (p{t) for all t e T. A multifunction q> is measur able if {teT: (p(t)r\G for each open G < = Y (such q is called weakly measurable by Himmelberg [10