On a Spector ultrapower of the Solovay model Vladimir Kanovei, Michiel van Lambalgen We prove that a Spector-like ultrapower extension N of a countable Solovay model M (where all sets of reals are Lebesgue measurable) is equal to the set of all sets constructible from reals in a generic extension M[\alpha] where \alpha is a random real over M. The proof involves an almost everywhere uniformization theorem in the Solovay model.