Common formulations of the general measurement postulate of quantum mechanics have two parts, one establishing the probability of the outcome $m$ corresponding to an observable $M_m$ for a system in an initial state $\left|\psi\right>$:
$$\left|\left|\psi_m\right>\right|^2=p(m) $$
where $\left|\psi_m\right>\equiv M_m\left|\psi\right>$; and a second part specifying the post measurement state of the system given that $m$ is observed.
But is the seond part necessary? Isn't it a consequence of the first part?
The first part establishes a relationship between component norms and probabilities of the form
$$\Vert\left| \psi\right>\Vert_E=\sqrt{\mathbb{P}(E)}$$
where $\Vert\left| \psi\right>\Vert_E$ is the norm of the component of a system $\left| \psi\right>$ corresponding the the occurrence of event $E$, specifically, the postulate states that
$$\Vert\left| \psi\right>\Vert_{O=m\wedge M=M_m}=\sqrt{\mathbb{P}(O=m\wedge M=\psi_m)}$$
where $O$ is the observed outcome and $M$ is the applied observable.
Is this correspondence between norms and probability a general one so that
$$\Vert\left| \psi\right>\Vert_{M=M_m\vert O=m}=\frac{\Vert\left| \psi\right>\Vert_{O=m\wedge M=M_m}}{\sqrt{\mathbb{P}(O=m)}}=\frac{\Vert\left| \psi_m\right>\Vert}{\sqrt{p(m)}}$$
and the second part of the postulate thus follows from the first part (keeping in mind that, by construction, the event $O=m'\wedge M=M_{m''}$ cannot occur for $m'\neq m''$)?
If so, isn't the second part of the postulate entirely unnecessary: establishing this "conditional amplitude" is its only content?
