In this work the problem of designing a state estimator for completely or partially observable continuous nonlinear plants with discrete measurements is addressed. The combination of a geometric approach with a stability analysis yields an estimator design methodology with a nonlinear detectability condition susceptible of testing, a systematic estimator construction, a robust convergence criterion coupled with a simple tuning scheme, as well as a rationale to explain the interplay between sampling time, estimator gains, and estimator functioning. Comparing with the continuous measurement case where the convergence is attained by tuning the gain above a low limit, in the discrete measurement case the loss of information due to the measurement sampling increases the size of the lower gain limit, and imposes sampling time and high gain limits. The proposed methodology is applied to address the estimation problem of a class of solution homopolymerization reactors, and is tested with a methyl-methacrylate polymerization run taken from a previous extended Kalman filter implementation study with experimental data.