  Using the renormalized-ring-diagram approximation, we study the
compressibility of the interacting electrons in bilayer graphene. The
compressibility is equivalent to the spin susceptibility apart from a constant
factor. The chemical potential and the compressibility of the electrons can be
significantly altered by an energy gap (tunable by external gate voltages)
between the valence and conduction bands. For zero gap and a typical finite gap
in the experiments, we show both systems are stable.
