  Microwave irradiation of a two-dimensional electron gas (2DEG) produces a
non-equilibrium distribution of electrons, and leads to oscillations in the
dissipative part of the conductivity. We show that the same non-equilibrium
electron distribution induces strong oscillations in the 2DEG compressibility
measured by local probes. Local measurements of the compressibility are
expected to provide information about the domain structure of the zero
resistance state of a 2DEG under microwave radiation.
