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(As far as im aware) No exact measurement of position is or ever will be possible. Suppose we made a measurement of the position of a particle using an instrument which measures the particle to be within a range of about 1fm. Would the wavefunction collapse to a wavefunction equal to zero outside of this region? Would a delta function be a suitable eigenstate of the position operator for a thoeretically ideal measurement? Or would the position operator have similar to eignestates of the momentum operator which are unnormalisable?

Adrien Amour
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    This answer might be of help. The idea is that you can construct a position-like operator that is degenerate in a subinterval of the real line. When you measure an operator that is degenerate, the post-measurement state is the projection of the pre-measurement state onto the degenerate subspace corresponding to the measurement result. – march Oct 08 '21 at 20:12
  • The link in @march's comment gives a practical answer. Remember, though, that any real measurement uses equipment made of molecules, connected to recording devices made of molecules, an so on. If you use a model that includes all that stuff as part of the quantum system (which is too hard, but we could do it in principle), then you can apply the projection postulate to a naturally-discrete observable somewhere downstream in that cascade of processes -- say a computer screen -- instead of applying it to the position operator itself. Then you don't need to discretize the position operator. – Chiral Anomaly Oct 08 '21 at 21:36

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