  The physical meaning of weak values and measurements can be completely
understood with Born rule and the general probability theory. It is known that
the weak value of an observable $\hat A$ with post-selection $\langle F|$ may
be out of the eigenvalue range of $\hat A$. This is because the weak value of
$\hat A$ with the post-selection is, in general, not the expectation value of
$\hat A$, but the expectation value of $\hat A| F\rangle\langle F|$ boosted by
the post-selection.
