Luzin measurability of Carathéodory type mappings

A topological space X is said to have the Scorza-Dragoni property if the following property holds: For every metric space Y and every Radon measure space ( T , μ ) , any Carathéodory function f : T × X → Y is Luzin measurable, i.e., given ε > 0 , there is a compact set K in T with μ ( T ∖ K ) ⩽ ε such that the mapping f : K × X → Y is continuous. We present a selection of spaces without the Scorza-Dragoni property, among which there are first countable hereditarily separable and hereditarily Lindelöf compact spaces, separable Moore spaces and even countable k-spaces. In the positive direction, it is shown that every space which is an ℵ 0 -space and k R -space has the Scorza-Dragoni property. We also prove that every separately continuous mapping f : T × X → Y , where Y is a metric space, is Luzin measurable, provided the space X is strongly functionally generated by a countable collection of its bounded subsets. If Martinʹs Axiom is assumed then all metric spaces of density less than c , and all pseudocompact spaces of cardinality less than c , have the Scorza-Dragoni property with respect to every separable Radon measure μ. Finally, the class of countable spaces with the Scorza-Dragoni property is closely examined.