We consider the problem of global stabilization of an unstable bioreactor model (e.g., for anaerobic digestion), when the measurements are discrete and in finite number ("quantized"), with control of the dilution rate. The measurements define regions in the state space, and they can be perfect or uncertain (i.e., without or with overlaps). We show that, under appropriate assumptions, a quantized control may lead to global stabilization: trajectories have to follow some transitions between the regions, until the final region where they converge toward the reference equilibrium. On the boundary between regions, the solutions are defined as a Filippov differential inclusion. If the assumptions are not fulfilled, sliding modes may appear, and the transition graphs are not deterministic.