Is there a reason eigenvalues of operators correspond to measurements?

[gsingh2011]

Given a wave function $\Psi$ which is an eigenstate of a Hermitian operator $\hat{Q}$, we can measure a definite value of the observable corresponding to $\hat{Q}$, and the value of this observable is the eigenvalue $Q$ of the eigenstate
$$
\hat{Q}\Psi = Q\Psi
$$
My question is whether it's a postulate of quantum mechanics that the eigenvalue of the eigenstate corresponds to the value we measure, or is there a more fundamental reason/proof for this being the case?

 

[PeroK]

It's a postulate.

 

[atyy]

The eigenvalues are labels for the outcomes. In general, an eigenvalue need not be the outcome itself.

There are also more general measurements (called POVMs) than projective measurements, and the labels here are not necessarily eigenvalues: https://arxiv.org/abs/0706.3526

One way to see that the eigenvalue is just a label for an outcome, and not necessarily literally the outcome itself is to use the indirect measurement formalism: https://arxiv.org/abs/1110.6815