Distances from selectors to spaces of Baire one functions

Given a metric space X   and a Banach space (E,‖⋅‖)(E,‖⋅‖) we study distances from the set of selectors Sel(F)Sel(F) of a set-valued map F:X→P(E) to the space B1(X,E)B1(X,E) of Baire one functions from X into E. For this we introduce the d-τ  -semioscillation of a set-valued map with values in a topological space (Y,τ)(Y,τ) also endowed with a metric d. Being more precise we obtain thatd(Sel(F),B1(X,E))⩽2oscw∗(F), where oscw∗(F) is the ‖⋅‖‖⋅‖-w-semioscillation of F. In particular, when F   takes closed values and oscw∗(F)=0 we get that then F has a Baire one selector: we point out that if F   is weakly upper semicontinuous then oscw∗(F)=0 and therefore our results strengthen a Srivatsa selection theorem when F takes closed set. We also obtain similar results when τ is the topology of convergence on some boundary B or τ   is the w∗w∗ topology of a bidual Banach space.