On Genericity and Complements of Measure Zero Sets in Function Spaces

Abstract

Generic properties of function spaces have been of particular interest in dynamical systems and singularity theory. The underlying assumption has been that the complement of a dense ${G_\delta }$ set is sparse enough to be considered unlikely. Nevertheless, in infinite dimensional spaces, even dense ${G_\delta }$’s may have measure zero. Since there is no one canonical measure on an infinite dimensional Fréchet space, notions of measure zero have not often been considered. Here we use a notion of Haar measure zero on abelian Polish groups due to Christensen [1]. We show that those sections of a finite dimensional vector bundle over a compact manifold whose jets are transverse to a submanifold of the jet bundle are complements of sets of Haar measure zero.