How will the wavefunction of a QM particle collapse after a measurement of position depending on the uncertainty of the measurement?

Question

(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?