  We present measurements of the electronic compressibility, $K$, of bilayer
graphene in both zero and finite magnetic fields up to 14 T, and as a function
of both the carrier density and electric field perpendicular to the graphene
sheet. The low energy hyperbolic band structure of bilayer graphene is clearly
revealed in the data, as well as a sizable asymmetry between the conduction and
valence bands. A sharp increase in $K^{-1}$ near zero density is observed for
increasing electric field strength, signaling the controlled opening of a gap
between these bands. At high magnetic fields, broad Landau level (LL)
oscillations are observed, directly revealing the doubled degeneracy of the
lowest LL and allowing for a determination of the disorder broadening of the
levels.
