Random gaps

Abstract

It is proved that there exists an (omega-1,omega-1) Souslin gap in the Boolean algebra (L(nu)/Fin,subseteq^*_ae) for every nonseparable measure nu. Thus a Souslin, also known as destructible, (omega-1,omega-1) gap in P(N)/Fin can always be constructed from uncountably many random reals. We explain how to obtain the corresponding conclusion from the hypothesis that Lebesgue measure can be extended to all subsets of the real line (RVM).