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 δ-finite measure space (X, A, μ) that covers μ-a.e. x ∈ X infinitely many times, then there exists a sequence of integers ni of density zero so that Ani still covers μ-a.e. x ∈ X infinitely 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 random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas. [ABSTRACT FROM AUTHOR]