>>263 補足

ところで、Ｔさんは
>>263 のSergiu Hart氏の PUZZLESのページ ”Choice Games”のＰＤＦで

最後に”Remark. When the number of boxes is finite Player 1 can guarantee a win
with probability 1 in game1, and with probability 9/10 in game2, by choosing
the xi independently and uniformly on [0, 1] and {0, 1, ..., 9}, respectively.”

とあることに気付いているのだろうか？

>>293のように、game1とgame2の差は、非加算と可算の違いだと
それで、game2は可算で、”here we do not use the Axiom of Choice.”だから、非可測ではない

で、上記Remarkのように、"When the number of boxes is finite"で
game1：Player 1 can guarantee a win with probability 1
game2：Player 1 can guarantee a win with probability 9/10