  We have measured the local electronic compressibility of a two-dimensional
hole gas as it crosses the B=0 Metal-Insulator Transition. In the metallic
phase, the compressibility follows the mean-field Hartree-Fock (HF) theory and
is found to be spatially homogeneous. In the insulating phase it deviates by
more than an order of magnitude from the HF predictions and is spatially
inhomogeneous. The crossover density between the two types of behavior, agrees
quantitatively with the transport critical density, suggesting that the system
undergoes a thermodynamic change at the transition.
