Alan Turing made the following claim: "It is easy to show using standard theory that if a system starts in an eigenstate of some observable, and measurements are made of that observable N times a second, then, even if the state is not a stationary one, the probability that the system will be in the same state after, say, 1 second, tends to one as N tends to infinity; i.e. that continual observation will prevent motion." But is he actually right about every single possible case? E.g. what if one continuously measures the position of a particle, can it move?