On a Spector Ultrapower of the Solovay Model
Vladimir Kanovei, Michiel van Lambalgen

Abstract:
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.