On Filling Families of Finite Subsets of the Cantor Set

Let ε > 0 and F be a family of finite subsets of the Cantor set C. Following D. H. Fremlin, we say that F is ε-filling over C if F is hereditary and for every F ⊆ C finite there exists G ⊆ F such that G ∈ F and |G| ≥ ε|F |. We show that if F is ε-filling over C and C-measurable in [C], then for every P ⊆ C perfect there exists Q ⊆ P perfect with [Q] ⊆ F . A similar result for weaker versions of density is also obtained.