It cannot be a Dirac delta as this is not a continuous normalisable function.

Further to the answer written by John Rennie How does wave function collapse when I measure position?

We may assume that the collapse is not perfect. In this case, how is the quantum formalism applied to calculate the probability of this measurement? One solution to this is to discretise the continuous operator as a function of position. The eigenstates are then degenerate and we will have a non zero probability of measurement. However, is it necessarily true that the resulting collapse will be a projection of the wavefunction? Can it not be 'reset' to a simpler distribution, for example Gaussian?