We study dynamics of the measurement process in quantum dot systems, where a particular state out of coherent superposition is observed. The ballistic point-contact placed near one of the dots is taken as a noninvasive detector. We demonstrate that the measurement process is fully described by the quantum rate equations applied to the whole system. These equations clearly reproduce the collapse of the density-matrix into the statistical mixture in the course of the measurement process. The corresponding dephasing width is uniquely defined. We show that the continuous observation of one of the states in a coherent superposition may accelerate decay from this state. This contradicts to a widespread viewpoint that the transitions between quantum states always slow down in the course of continuous measurement (the quantum Zeno effect).