open dense and prevalent subset.

It follows directly from Theorem \[Th1\] and the fact that the subspace of $\mathcal{H}^\alpha_L(\omega)$ for which  holds is open, dense and prevalent. Indeed, prevalence follows form the fact that  holds for $H_Q$ for a set of full Lebesgue measure of $Q \in P_2(n,m_0)$ (see [@HK10] for a survey on the notion of prevalence). Openness follows from the fact that  is an open condition, and density then follows from prevalence.

Now let us consider our second setting, that is we consider a Hamiltonian as in .

We say that a completely integrable Hamiltonian $h\in G^\alpha_L(\bar{D})$ is [*KAM doubly exponentially stable with exponent $u$*]{} if for any $f \in G^\alpha_L(\T^n \times \bar{D})$, and for any $\gamma>0$, the following holds: there exists $\varepsilon_*>0$ such that for any $\varepsilon\leq \varepsilon_*$, there exists, for the Hamiltonian flow of $H_\varepsilon$ as in , a set of invariant Lagrangian Diophantine tori $\mathcal{K}' \subseteq \T^n \times \bar{D}$ whose complement in $\T^n \times \bar{D}$ has a measure of order $\gamma$, and such that every torus in $\mathcal{K}' $ is doubly exponentially stable with exponent $u$.

Given a smooth function $h$ defined on the closure $\bar{D}$ of some open bounded domain $D \subseteq {\mathbb{R}}^n$, for 