Local openness: Characterization of a robustness concept for generic sets in function spaces

Abstract

This paper introduces a novel robustness concept for generic sets in function spaces. As motivation, I provide an economically relevant example of a functional property, where a subset of the domain with arbitrary small (Lebesgue) measure marks the difference between the functional property being strongly negligible (nowhere dense) or non-negligible (open). I define a functional property as locally open at some point of the domain if there exists a closed ball around the point such that the functional property is open in the continuous functions on the domain restricted to the ball. A functional property is open if and only if it is locally open everywhere on the domain. Further, a subset of the domain with arbitrary small measure marks the difference between an open and nowhere dense property if and only if the functional property is locally open almost everywhere on the domain.