A covering theorem and the random-indestructibility of the density zero ideal

The main goal of this note is to prove the following theorem. If An is a sequence of measurable sets in a σnite measure space (X,A, μ) that covers μ-a.e. x ∈ X in nitely many times, then there exists a sequence of integers ni of density zero so that Ani still covers μ-a.e. x ∈ X in nitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is randomindestructible, that is, random forcing does not add a co-in nite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas.