A Covering Theorem and the Random-Indestructibility of the Density Zero Ideal

Abstract

The main goal of this note is to prove the following theorem. If \(A_n\) is a sequence of measurable sets in a \(\sigma\)-finite measure space \((X, \mathcal{A}, \mu)\) that covers \(\mu\)-a.e. \(x \in X\) infinitely many times, then there exists a sequence of integers \(n_i\) of density zero so that \(A_{n_i}\) still covers \(\mu\)-a.e. \(x \in X\) 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.