Baire class $1$ selectors for upper semicontinuous set-valued maps

Abstract:

Let $T$ be a metric space and $X$ a Banach space. Let $F:T \to X$ be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition: $\{ t \in T:F(t) \cap C \ne \emptyset \}$ is closed for each weakly closed set $C$ in $X$. Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for $F$. We prove a statement of similar precision and generality when $X$ is a metric space.