Consider a stochastic process $X_{t}$ taking values in the set $\{0,1\}$ according to the probability measure $\mu$. Let $$Y_{t} = \mu\left(\limsup_{T \rightarrow \infty}\frac{1}{T}\sum_{t = 0}^{T - 1}X_{t} \geq \frac{1}{2}\mid X_{0},\dots,X_{t - 1}\right).$$ We know that $$\mu(X_{t} = 1 \mid Y_{t} < \epsilon) > 1 - \epsilon.$$ Show that $$Y_{t} = 1\text{ for every }t,\,\mu\text{-almost surely.}$$

We expect this has to do with Levy's zero-one law, but we are unable to prove the conjecture. Thank you so much for your comments!