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The fermionic Hubbard model plays a fundamental role in the description of strongly correlated materials. We have realized this Hamiltonian in a repulsively interacting spin mixture of ultracold 40K atoms in a three-dimensional (3D) optical lattice. Using in situ imaging and independent control of external confinement and lattice depth, we were able to directly measure the compressibility of the quantum gas in the trap. Together with a comparison to ab initio dynamical mean field theory calculations, we show how the system evolves for increasing confinement from a compressible dilute metal over a strongly interacting Fermi liquid into a band-insulating state. For strong interactions, we find evidence for an emergent incompressible Mott insulating phase. This demonstrates the potential to model interacting condensed-matter systems using ultracold fermionic atoms.