Density zero slaloms Original Research Article

We construct a View the MathML source set G⊆ωω×2ω with null vertical sections such that each perfect set P⊆2ω meets almost all vertical sections of G in the following sense: we can define from P subsets S(i)(i∈ω) of ω of density zero such that whenever the section determined by x∈ωω does not meet P, then x(i)∈S(i) for all but finitely many i. This generalizes theorems of Mokobodzki (Ensembles à coupes dénombrables, Séminaire de Probabilités, Université de Srasbourg 1977/78) and Brendle et al. (Ann. Pure Appl. Logic 59 (1992) 185–199).