I have defined a set of quantum measurement operators of the form $$A_C = \sum _M \sqrt{Pr(M|C)} |M \rangle \langle M|$$ where $Pr(M|C)$ is the Poisson distribution $$Pr(M|C) = \frac{e^{-C}C^M}{M!}~~~~M = 0,1,2...$$ with mean $C$ and $|M \rangle$ are eigenstates of some observable and where there are finite members of $\{ |M \rangle \}_M$. I need the set of operators to satisfy the completeness condition $$\sum_{C}A_C^{\dagger}A_C = I.$$ My first attempt at this was to show that this is already satisfied: $$\sum_{C}A_{C}^{\dagger}A_C = \sum_{C}\sum_{M}Pr(M|C)|M \rangle \langle M| = \sum_{M}\sum_{C}Pr(M|C)|M \rangle \langle M| = \sum_M |M \rangle \langle M | = I$$ but having checked, I don't think the second last equation is necessarily valid for Poisson distributions. Can anyone see how I can adapt this so that the completeness relation is satisfied?
Thanks for any assistance.