Suppose we have a quantum system that is in a certain initial state, together with a measuring device capable of limited precision measurement of some observable. After the measurement is complete, we see that the result of the measurement belongs to some Borel subset $\Omega$ of $\mathbb{R}$. Does this mean that the original state has (after the measurement) collapsed into a state obtained by projecting the original state with the projector associated to $\Omega$? What if someone else uses a higher precision device?
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