Suppose that I accept that there is wave function collapse in quantum mechanics, and that the probabilities associated with each orthogonal subspace are a function of the wave function $\psi$ before the collapse.
I've seen some references that claim that in this case, Gleason's theorem implies that the probabilities are given by Born's rule, that is, by the squares of the absolute values of the amplitudes of $\psi$ (here is one such reference).
Loosely speaking, Gleason's theorem asserts that for any probability measure $\mu$ on a Hilbert space $\mathcal{H}$ (I mean, in the quantum sense, where $\mu$ is defined on subspaces of $\mathcal{H}$, and is additive under the sum of orthogonal subspaces) there is a state $\phi\in\mathcal{H}$ (more correctly: a density matrix) such that $\mu$ can be expressed by Born's rule using $\phi$.
I'm trying to understand how Gleason's theorem implies Born's rule. In other words, why is the $\phi$ in the theorem identical to $\psi$? Would there be any contradiction if for a state $\psi$ the probabilities were given by the forth powers of the amplitudes of $\psi$? I understand that in this case, $\psi\neq\phi$, but is there any problem with this?
Here is a related question, but it seems to me that it discusses a different issue - of how probabilities emerge in the many worlds interpretation.