Suppose that $A$ is a non-determined set in Baire space (although perhaps my argument is a little easier to see for Cantor space, using the usual coin-flipping measure), and we interpret player I as trying to play into $A$. Let's build a new set $B$ from $A$ by inserting a $0$ into every other play by player I, and ignoring the next move by player II. That is, we insert a round of dummy moves between each real round of play, insisting in the dummy moves that I play $0$ and II can play anything. Thus, I wins the game for B by making sure to play a $0$ in the dummy rounds, and otherwise paying attention only to the real moves of player II. Player II wins by ignoring the dummy moves, and otherwise playing as in $A$. Thus, the game $B$ is not determined, since from a strategy for $B$ for either player we could make a strategy for $A$ by inventing appropriate dummy moves. But meanwhile, because of all those inserted $0$s, every sequence in $B$ has every forth digit $0$, and so the measure of $B$ is zero, and so $B$ is a measurable set. 

This argument does not seem to use AC, but only the existence of a non-determined set.