For a field K and a positive integer e let N e (K) be the set of all e-tuples σ = (σ 1, ..., σ e)εG(K) e that generate a selfnormalizer closed subgroup of G(K). Chatzidakis proved, that if K is Hilbertian and countable, then N e (K) has Haar measure 1. If K is Hilbertian and uncountable, this need not be the case. Indeed, we prove that if K 0 is a field of characteristic 0 that contains all roots of unity, T is a set of cardinality א1 which is algebraically independent over K 0 and K =K 0(T), then neither N e (K) nor its complement contain a set of positive measure. In particular N e (K) is a nonmeasurable set.