  Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the
cotangent space $T^*M$, strictly convex and superlinear in the momentum
variables, we prove uniqueness of certain ergodic invariant Lagrangian graphs
within a given homology or cohomology class. In particular, in the context of
quasi-integrable Hamiltonian systems, our result implies global uniqueness of
Lagrangian KAM tori with rotation vector $\rho$. This result extends
generically to the $C^0$-closure of KAM tori.
