$Q \in G$, the set $KS(Q)$ has full Lebesgue measure, that is $h_Q$ satisfies both conditions  and  at almost all points $I \in \bar{D}$. In particular, for any $Q \in G$ and for any $\gamma>0$, $h_Q$ satisfies both conditions  and  on a compact subset of $\bar{D}$ whose complement in $\bar{D}$ has a measure less than $\gamma$. This shows that $G$ is contained in $G_\gamma$ (in fact, $G$ is contained in the intersection of the $G_\gamma$ over $\gamma>0$), and therefore $G_\gamma$ has full Lebesgue measure.

Plan of the proofs
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The proofs of Theorem \[Th1\] and Theorem  \[Th2\], although independent from each other, will follow the same path, which was the one also taken in [@BFN15]. Let us first start with the case of a single Lagrangian invariant Diophantine torus.

The first step consists of constructing a Birkhoff normal form with an exponentially small remainder. In the analytic case, even though the Birkhoff formal series is expected to be divergent in general, its coefficients have a moderate growth of Gevrey type, so that on a ball of radius $r$, if one truncates the series at a large order $m \sim r^{-a}$, the non-integrable part in the normal form becomes exponentially small 