  We analyse the measure of the regularized matrix model of the supersymmetric
potential valleys, $\Omega$, of the Hamiltonian of non zero modes of
supermembrane theory. This is the same as the Hamiltonian of the BFSS matrix
model. We find sufficient conditions for this measure to be finite, in terms
the spacetime dimension. For $SU(2)$ we show that the measure of $\Omega$ is
finite for the regularized supermembrane matrix model when the transverse
dimensions in the light cone gauge $\mathrm{D}\geq 5$. This covers the
important case of seven and eleven dimensional supermembrane theories, and
implies the compact embedding of the Sobolev space $H^{1,2}(\Omega)$ onto
$L^2(\Omega)$. The latter is a main step towards the confirmation of the
existence and uniqueness of ground state solutions of the outer Dirichlet
problem for the Hamiltonian of the $SU(N)$ regularized $\mathrm{D}=11$
supermembrane, and might eventually allow patching with the inner solutions.
