The rational measure game

Work in ZF, with $\mathcal{C}$ as in Cantor sequence space. For $E\subseteq \mathcal{C}$ and rational $0<v\le 1$, the rational measure game starts with current bound v. On each I turn the move is a rational pair $(h_0,h_1)\in [0,1{]}^2$ satisfying $(h_0+h_1)\mathrm{/}2\ge v_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$. II then chooses a bit e with $h_e>0$, and the next current bound is h_e. I wins exactly when II's infinite bit sequence belongs to E. Strategies remember the full sequence of pairs and bits, according to Gale–Stewart games and strategies.

Using $\mathbb{Q}$ is countably infinite, fix an enumeration of rational numbers and code pairs by $⟨i,j⟩=(i+j)(i+j+1)\mathrm{/}2+j$. This encodes I moves by naturals; II's legal bit codes are zero and one. The first illegal move loses immediately, regardless of later moves. Thus the payoff on ${\mathbb{N}}^{\mathbb{N}}$ is the union of plays with first illegal II move and wholly legal plays with bit outcome in E.

Every legal position has a legal continuation. I can choose (1,1), and every legal pair has a positive coordinate because the current bound is positive. II can take the least positive coordinate. The next bound remains rational in (0,1], including when it equals one. Thus legal full histories exist by least-code recursion; zero selected coordinates are illegal. A winning coded strategy cannot prescribe a first illegal move on a legal consistent history, since the opponent can always continue legally. Restrict it there and fill inconsistent legal histories with least legal defaults to obtain a legal winning strategy. This definition assumes neither AD nor AC, and asserts no measure exists.