  In this paper we prove that a strictly convex Alexandrov solution u of the
Monge-Amp\`ere equation, with right hand side bounded away from zero and
infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher
integrability a-priori estimates for $D^2 u$, namely $D^2 u \in L\log^k L$ for
any $k\in \mathbb N$.
