  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.
