  We describe an experimental technique to measure the chemical potential,
$\mu$, in atomically thin layered materials with high sensitivity and in the
static limit. We apply the technique to a high quality graphene monolayer to
map out the evolution of $\mu$ with carrier density throughout the N=0 and N=1
Landau levels at high magnetic field. By integrating $\mu$ over filling factor,
$\nu$, we obtain the ground state energy per particle, which can be directly
compared with numerical calculations. In the N=0 Landau level, our data show
exceptional agreement with numerical calculations over the whole Landau level
without adjustable parameters, as long as the screening of the Coulomb
interaction by the filled Landau levels is accounted for. In the N=1 Landau
level, comparison between experimental and numerical data reveals the
importance of valley anisotropic interactions and the presence of
valley-textured electron solids near odd filling.
