Random gaps
James Hirschorn · Transactions of the American Mathematical Society · 2008
It is proved that there exists an ( ω 1 , ω 1 ) (\omega _1,\omega _1) Souslin gap in the Boolean algebra ( L 0 ( ν ) / Fin , ⊆ ae ∗ ) (L^0( u )/\operatorname {Fin}, \subseteq _{\operatorname {ae}}^*) for every nonseparable measure ν u . Thus a Souslin, also known as destructible, ( ω 1 , ω 1 ) (\omega _1,\omega _1) gap in P ( N ) / Fin \mathcal {P}(\mathbb {N})/ \operatorname {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).