  We use Naimark's dilation theorem in order to characterize the joint
measurability of two POVMs. Then, we analyze the joint measurability of two
commutative POVMs $F_1$ and $F_2$ which are the smearing of two self-adjoint
operators $A_1$ and $A_2$ respectively. We prove that the compatibility of
$F_1$ and $F_2$ is connected to the existence of two compatible self-adjoint
dilations $A_1^+$ and $A_2^+$ of $A_1$ and $A_2$ respectively. As a corollary
we prove that each couple of self-adjoint operators can be dilated to a couple
of compatible self-adjoint operators. Next, we analyze the joint measurability
of the unsharp position and momentum observables and show that it provides a
master example of the scheme we propose. Finally, we give a sufficient
condition for the compatibility of two effects.
