Say I have two commuting operators $A$ and $B$, with joint eigenvectors $|n\rangle$.
Say I have a state $S = \sum_n a_n | n \rangle$. If I measure $A$ first, I pick out say an eigenstate $|k\rangle$ and then measure $B$ which just gives me $|k\rangle$ again. But if I measure $B$ first, it might pick out a different eigenstate, as $|m\rangle$ and then measuring $A$ gives $|m\rangle$. So it seems measurements don't necessarily commute unless acting on a joint eigenstate. So my question is whether this statement is true and statements about measurements of commuting observable being order independent is false.