  Following D. Sobota we call a family $\mathcal F$ of infinite subsets of
$\mathbb N$ a Rosenthal family if it can replace the family of all infinite
subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of
measures on pairwise disjoint sets. We resolve two problems on Rosenthal
families: every ultrafilter is a Rosenthal family and the minimal size of a
Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This
is achieved through analyzing nowhere reaping families of subsets of $\mathbb
N$ and through applying a paving lemma which is a consequence of a paving lemma
concerning linear operators on $\ell_1^n$ due to Bourgain. We use connections
of the above results with free set results for functions on $\mathbb N$ and
with linear operators on $c_0$ to determine the values of several other derived
cardinal invariants.
