A selection theorem for multifunctions

Abstract:

In this paper the following theorem is proved. X is any set, H is a family of subsets of X which is $\lambda$-additive, $\lambda$-multiplicative and satisfies the $\lambda$-WRP for some cardinal $\lambda > {\aleph _0}$. Suppose Y is a regular Hausdorff space of topological weight $\leqslant \lambda$ such that given any family of open sets, there is a subfamily of cardinality $< \lambda$ with the same union. Let $F:X \to {\mathbf {C}}(Y)$, where ${\mathbf {C}}(Y)$ is the family of nonempty compact subsets of Y, satisfy $\{ x:F(x) \cap C \ne \emptyset \} \in {\mathbf {H}}$ for any closed subset C of Y. Then F admits a ${({\mathbf {H}} \cap {{\mathbf {H}}^c})_\lambda }$ measurable selector.