  In this paper we prove that $M^p_\Lambda$ is almost isometric to $\ell^p$ in
the canonical way when $\Lambda$ is lacunary with a large ratio. On the other
hand, our approach can be used to study also the Carleson measures for M\"untz
spaces $M^p_\Lambda$ when $\Lambda$ is lacunary. We give some necessary and
some sufficient conditions to ensure that a Carleson embedding is bounded or
compact. In the hilbertian case, the membership to Schatten classes is also
studied. When $\Lambda$ behaves like a geometric sequence the results are
sharp, and we get some characterizations.
