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Suppose I have some density operator, expressed in the position eigenbasis:

$$\rho =\int p(x)|x\rangle\langle x | dx,$$ where $p(x)$ is some probability density.

Then a projective position measurement of $\rho$ should yield (up to a normalization factor):

\begin{align}\rho' &= \int\int dx dy \ p(x)\ |y\rangle\langle y|x\rangle\langle x| y \rangle \langle y| \\ \end{align}

Now evaluating the above inner products yields two Dirac deltas, which makes $\rho \rightarrow \infty$, which is clearly incorrect. I know the answer should simply be $\rho'=\rho$, however I am unable to show this. Does anyone know how the trick to solving the above integral?

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