constants characterizing the conditions  and . The latter uniform constants can be obtained using the compactness of $\bar{D}$ and the fact that $N(n,2)$ and $SS(n,m_0)$ are open.

As before, it remains to explain in which sense the conditions  and  are generic. In fact we have the following lemma.

\[lemma.fub\] Fix $\gamma>0$. Given any $h \in G^\alpha_L(\bar{D})$, for an open set of full Lebesgue measure of $Q \in P_2(n,m_0)$, the modified integrable Hamiltonian $h_Q \in G^\alpha_L(\bar{D})$ defined by $h_Q(I):=h(I)+Q(I)$ satisfies  and  on a compact subset of $\bar{D}$ whose complement in $\bar{D}$ has a measure less than $\gamma$.

It immediately follows that the set of $Q  \in P_2(n,m_0)$ such that for any $\gamma>0$, $h_Q$ satisfies  and  (with constants that depend of course on $\gamma$) on a compact subset of $\bar{D}$ whose complement in $\bar{D}$ has measure less than $\gamma$, contains a residual and prevalent set. Hence we get the following corollary of Theorem \[Th2\] and Lemma \[lemma.fub\].

\[coro2\] For a residual and prevalent set of integrable Hamiltonians $h\in  G^\alpha_L(\bar{D})$, $h$ is [*KAM doubly exponentially stable with exponent $u$*]{} for any $u<\frac{1}{{\alpha}n}$.

Let us now give the proof of Lemma \[lemma.fub\].

Let us denote by $G_\gamma$ the set of $Q \in P_2(n,m_0)$ such 