that $h_Q$ satisfies both conditions  and  on a compact subset of $\bar{D}$ whose complement in $\bar{D}$ has a measure less than $\gamma$. What we need to prove is that $G_\gamma$ is both open and has full Lebesgue measure.

Since the sets $N(n,2)$ and $SS(n,m_0)$ are open, by a compactness argument the set $G_\gamma$ is open, so it remains to prove that $G_\gamma$ has full Lebesgue measure.

Let us define the subset $KS \subset \bar{D} \times P_2(n,m_0)$ consisting of couples $(I,Q)$ for which the modified Hamiltonian $h_Q$ satisfies both conditions  and  at the point $I$. Let us also define the sections $$KS(I):=\{Q \in P_2(n,m_0) \; | \; (I,Q) \in KS \}, \quad KS(Q):=\{I \in \bar{D} \; | \; (I,Q) \in KS\} .$$ For any $I \in \bar{D}$, we claim that the set $KS(I)$ has full Lebesgue measure. Indeed, it is obvious that the set of $Q  \in P_2(n,m_0)$ for which condition  is satisfied at $I$ has full Lebesgue measure while the fact that the set of $Q  \in P_2(n,m_0)$ for which condition  is satisfied at $I$ has full Lebesgue measure follows from [@BFN15]. By Fubini’s theorem, there exists a subset $G \subset P_2(n,m_0)$ of full Lebesgue measure such that for any 