Hi!

Let X be the infinite coin toss space, ie X=\{0,1\}^\mathbb{N} and A_q:=\{\omega \in X: \lim_{n \to \infty} 1/n \sum_{k=1}^{n} \omega_k = q \}. I read that X-\bigcup_{q \in [0,1]} E_q is uncountable but i have absolutely no clue how to prove that. Does anyone have any ideas?

Best regards,
Banach