Let $ \ m_{*} \ $ be the Lebesgue Exterior measure on $ \ \mathbb{R}^{d} \ $ , where $ \ \mathbb{R}^{d} \ $ is the countable product of $ \ \mathbb{R} \ $.
Then compute
(i) $ \large \ m_{*} (\mathbb{Q}^d \cap [0,1]^d) \ $
Answer:
If $ \ d \ $ be finite , then $ Q^d \cap [0,1]^d \ $ will be countable and hence $ \ m_* (Q^d \cap [0,1]^d )=0 $
But if $ \ d \ $ be countably infinite , then $ \ Q^d \ $ may be uncountable and in that case I can't proceed in above way.
Help me out