  Consider a real--analytic nearly--integrable mechanical system with potential
$f$, namely, a Hamiltonian system, having a real-analytic Hamiltonian $$
H(y,x)=\frac12 | y |^2 +\e f(x)\ , $$
  $y,x$ being $n$--dimensional standard action--angle variables (and $|\cdot|$
the Euclidean norm). Then, for "general" potentials $f$'s and $\e$ small
enough, the Liouville measure of the complementary of invariant tori is smaller
than $\e|\ln \e|^a$ (for a suitable $a>0$).
