In this paper the following theorem is proved. X is any set, H is a family of subsets of X which is λ \lambda -additive, λ \lambda -multiplicative and satisfies the λ \lambda -WRP for some cardinal λ > ℵ 0 \lambda > {\aleph _0} . Suppose Y is a regular Hausdorff space of topological weight ⩽ λ \leqslant \lambda such that given any family of open sets, there is a subfamily of cardinality > λ > \lambda with the same union. Let F : X → C ( Y ) F:X \to {\mathbf {C}}(Y) , where C ( Y ) {\mathbf {C}}(Y) is the family of nonempty compact subsets of Y , satisfy { x : F ( x ) ∩ C ≠ ∅ } ∈ H \{ x:F(x) \cap C \ne \emptyset \} \in {\mathbf {H}} for any closed subset C of Y . Then F admits a ( H ∩ H c ) λ {({\mathbf {H}} \cap {{\mathbf {H}}^c})_\lambda } measurable selector.