Hi!

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

Best regards,

Banach