Special selectors of multivalued mappings

The following result related to the selection theorems due to Michael and to Kuratowski and Ryll-Nardzewski is given: Let X be a metric space with a sigma-finite regular measure, Y be a separable Banach space and Fcolon Xto Y be a measurable multifunction with closed values. Assume that B_0subset X is closed and has a base consisting of open-closed sets and B_1subset X is such that F(x) is convex for every xin B_1. If F is lower semicontinuous onbreak B_0cupoverline B_1, then it has a measurable selection which is continuous onbreak B_0cup B_1.