In this paper, an optimal measurement feedback control problem that yields an almost-dissipative closed loop system is considered. Using information state ideas and the definition of the optimal cost presented, a dynamic programming equation is derived. Incremental analysis yields a corresponding variational inequality (VI) which naturally generalizes the information state based partial differential equation (PDE) associated with measurement feedback nonlinear H-infinity control. In theory, this variational inequality can be used to synthesize an optimal measurement feedback controller which guarantees that the closed loop system almost satisfies a given dissipation property. This “almost-dissipation” property admits a weaker form of stability for the closed loop system, allowing presistence of excitation in the absence of disturbance inputs. Finally, certainty equivalence control is investigated as a special case of the results presented.