In many applications it is desired to limit the amount of sensor measurements for example to save energy. Developing control strategies that only perform measurements if necessary to achieve the required control performance is thus desirable. This work presents a combination of a robust predictive controller with self-triggered estimation for the control of linear, constrained systems subject to additive noise. We assume that for some sensors the maximum number of measurements within a specific time-span is limited, but the actual time-instants at which the measurements are taken can vary. We propose an output feedback control scheme, which optimizes the applied input as well as the measurement instances to improve closed loop performance. Conditions to robustly guarantee closed loop properties such as constraint satisfaction, recursive feasibility or stability are presented. The results are illustrated by an example.