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The main goal of this note is to prove the following theorem. If $A_n$ is asequence of measurable sets in a $\sigma$-finite measure space $(X,\mathcal{A}, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, thenthere 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 aprobabilistic construction.  As an application we give a simple direct proof of the known theorem that theideal 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 almostcontains every ground model density zero set. This answers a question of B.Farkas.

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