bhobba said:
Not quite.
There is a copy of you in all of them and each copy experiences a different outcome. What copy you are and what you experience is the same thing.
This is probably a bit too metaphysical, but if you split into two identical copies, one having experience A, and the other experience B, in order to be able to talk about 'you' being one of these copies (as opposed to the other), you have to introduce some means that picks out your 'you'-ness, i.e. that makes it so that your continuous experience is with copy B, rather than A, say. This is the route Albert and Loewer have considered (but rejected), and under such a constraint, it's indeed possible to make sense of probability in the 'MWI' (though of course it's not really the MWI anymore, but essentially a hidden-variable theory with an observer's mind being the hidden variable), by simply postulating that whatever this 'you'-index is, it 'jumps' stochastically into one of the two copies (as opposed to the other), thus providing a basis for a probabilistic interpretation. But I think that many worlds traditionally does not boil down to such a view; rather, one would typically hold that both copies are indeed equally much you, with no additional distinguishing features (though how this works is a bit of a subtle issue).
Anyway, it seems most people have made up their minds, but I've decided nevertheless to flesh out the reason why Gleason does not help in the MWI a little. I won't really be telling anybody anything new, but I think it helps looking at the story in a different way from how it's usually told.
Let's start classical. Consider a set of classical objects---marbles, say. This set is partitioned into subsets, according to some distinguishing characteristic of the marbles---say, their colour; say furthermore that we have four colours, red, green, blue, and pink. A subset of the total marble set corresponds to a proposition; whether or not a given marble belongs to that subset corresponds to the truth value of that proposition, i.e. 'the marble is red' is true if the marble belongs to the subset of red marbles.
Now, let's assume there's some fixed quantity of marbles---say 100---, and that in each subset, there's a fixed number of marbles, as well---50 red ones, 30 green ones, 19 blue ones and 1 pink marble. This gives us a way to associate a measure with the subsets---the set of red marbles gets measure 0.5, and so on, while the full set obviously gets measure 1. Now, we can make sense of the construction 'the probability that the marble has colour c is p'. If we don't know anything about the marble---draw it at random---, then, for instance, the probability that the marble is green is 30%. (This doesn't consider any subtleties about probability, and isn't meant to; even without the precise definitions, I suppose it's obvious to anybody that we can grasp the extension of the concept of probability.) If we know something---say, that the marble is neither green nor pink---we can adjust our probabilities accordingly. In fact, writing P_R for the proposition that the marble is red, and P_B for the proposition that it is blue, we might represent our knowledge of the situation by the quantity
r=p_1P_R+p_2P_B.
Now let's move over to quantum mechanics. Here, we don't have a set, but a Hilbert space, and we don't have subsets, but (closed) subspaces. And we also don't have propositions in the classical sense, but we can associate to every subspace a special operator, namely the projector on that subspace. So, we can kinda do the same thing we did before, and imagine the Hilbert space partitioned into subspaces, and any state belonging to a certain subspace has a certain property---assume there's four, and call them R, G, B, P. But here's the first problem: the counting measure doesn't really make sense anymore. Luckily, there's a way out: as Gleason showed, there's a unique measure attributable to closed subspaces, and it's given by the squared amplitudes (we don't need to get into any technicalities here, as most people are familiar with them anyway). So, we can play the same game as before---well, almost. We can play the same game
if we have some assurance that the state is in one of the subspaces we're considering, that is, determinately has any of the properties R, G, B, P. Then, we can proceed as before, and (again, given knowledge that the state does not have property G or P) write, for instance
\rho=|c_1|^2P_R+|c_2|^2P_B,
which, with P_R=|\psi_R\rangle\langle \psi_R| and P_B=|\psi_B\rangle\langle \psi_B|, corresponds to the proper mixture
\rho=|c_1|^2|\psi_R\rangle\langle \psi_R|+|c_2|^2|\psi_B\rangle\langle \psi_B|.
Up to this point, anything proceeds analogously to the classical case, thanks to Gleason. However, in general, we don't have the assurance that the state will have any of the properties determinately; in general, we're faced with a superposition such as
|\psi\rangle=c_1|\psi_R\rangle + c_2|\psi_B\rangle.
The question now is, how do we get from |\psi\rangle to \rho? And on the standard interpretation, the answer to this is the collapse; \rho is just the state after a measurement has occurred on |\psi\rangle, but before anybody had a look, so to speak. This is the key point: we must get to \rho before Gleason is of any help.
However, on the many worlds interpretation, no such mechanism is available. The state never collapses; we're left with |\psi\rangle at all times. Hence, Gleason doesn't say anything about the probabilities we ought to expect---in contrast to the case of the collapse. Appealing to Gleason's theorem, then, to the best of my ability to tell, is simply a nonstarter in the case of many worlds; it simply doesn't apply.
