= Solution
Introduce a <quantum ancilla> with basis $|i\rangle$ and define
$$
V|\psi\rangle=\sum_iM_i|\psi\rangle\otimes|i\rangle.
$$
The completeness relation gives $V^\dagger V=I$, so $V$ is an <isometry> and extends to a <unitary operator> on a sufficiently large system-plus-ancilla space. Prepare the ancilla in a fixed state, apply that unitary, and perform the projective measurement $\{I\otimes|i\rangle\langle i|\}$. Outcome $i$ has probability $\|M_i|\psi\rangle\|^2$ and leaves the system in the normalized state $M_i|\psi\rangle$. By linearity the same holds for mixed states, implementing the generalized measurement.
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