Assume we have a quantum system and its wavefunction $\psi$, but the wavefunction is not an eigenfunction of some operator (e.g. the z-component of the angular momentum $\hat{L_z}$).

In this case, how are we going to measure $L_z$?

Is it always the case that we can write $\psi$ as a sum of eigenfunctions of $\hat{L_z}$, and from that we extract an $L_z$ measurement?