Does the measurement take place when nobody observes it?

We consider {\em non-selective} continuous measurements of a particle tunneling to a reservoir of finite band-width ($\Lambda$). The particle is continuously monitored by frequent projective measurements ("quantum trajectory"), separated by a time-interval $\tau$. A simple analytical expression for the decay rate has been obtained. For Markovian reservoirs ($\Lambda\to\infty$), no effect of the measurements is found. Otherwise for a finite $\Lambda$, the decay rate always depends on the measurement time $\tau$. This result is compared with alternative calculations, with no intermediate measurements, but when the measurement device is included in the Schr\"odinger evolution. We found that the detector affects the system by the decoherence rate ($\Gamma_d$), related to the detector's signal. Although both treatments are different, the final results become very close for $\tau=2/\Gamma_d$. This $\tau$ corresponds to the minimal time for which the detector's signal can be distinguished by an "observer". This indicates a fundamental role of information in quantum motion and can be used for the extension of the quantum trajectory method for non-Markovian environments.