  We explore some variants of the Boman covering lemma, and their relationship
to the boundedness properties of the maximal operator. Let $1 < p < \infty$ and
let $q$ be its conjugate exponent. We prove that the strong type $(q,q)$ of the
uncentered maximal operator, by itself, implies certain generalizations of the
Boman covering lemma for the exponent $p$, and in turn, these generalizations
entail the weak type $(q,q)$ of the centered maximal operator. We show by
example that it is possible for the uncentered maximal operator to be unbounded
for all $1 < s < \infty$, while the conclusion of the lemma holds for every $1
< p < \infty$; thus, the latter condition is much weaker. Also, the boundedness
of the centered maximal operator entails weak versions of the lemma.
