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Basically the question is about three observables A,B,C. They obey the following rules:

[A,B]=0, [A,C]=0, but [B,C] not equal to 0.

So this is like the situation with the total squared angular momentum L^2 and the axial angular momenta Lx, Ly, Lz. Now I know that if the observables commute, they "share a complete set of eigenstates." So in this situation you can have eigenstates of A that are also eigenstates of B or C, but not both at the same time [yes?]. So you can't have a single quantum state that is an eigenstate of all three [yes?]. Now the question asks about the effects of these relations on sequential measurements of the observables. If you start in an A eigenstate, and measure B or C, could the state be left unchanged since they both "contribute" eigenstates to A? If you start in B or C and measure the other, will you get an even spread across the available eigenstates since they don't commute?

Or, am I drifting hopelessly in Hilbert Space??