I know that a game has imperfect information if there is an nonsingleton information set, and that defines a situation on which a player can't distinguish between the nodes in the information set. However, I'm wondering if, conceptually, it makes sense that the payoffs can be different, and still the player at the nonsingleton information set can't distinguish between the nodes. For example, consider an extensive form game with imperfect information like the following:
Which appears in Game Theory Online. There is an information set with two nodes for player 1 after player 2 choses A or B. Why player 1 can't distinguish at which node she is by the payoffs of, say $l$? If she recognizes that the payoff of $(0,0)$ corresponds for $l$ after $A$ and $(2,4)$ for $l$ after $B$, doesn't this disambiguate the situation?