Let $(Y_0,Y_1,\dots)$ be an $F_n$ martingale. Prove or disprove that the sets $\{\omega:\sup \{Y_n(\omega),n\ge 0\} < \infty\}$ and $\{\omega: \inf\{Y_n(\omega) , n\ge 0\}> -\infty\}$ coincide except on a null set.
I think this statement is not true, but I think it is really difficult to find a counter example. I tried some standard martingales, but could not conclude anything.
Let $Y_n = \sum_{i=1}^n \xi_i$ for independent random variables $\xi_i$ satisfying
$$\mathbb{P}(\xi_i=i^2) = \frac{1}{2i^2} \qquad \qquad \mathbb{P}(\xi_i=-1) = \frac{1}{2}.$$
Use the Borel Cantelli lemma to show that $Y_n \to - \infty$ almost surely. In particular,
$$\mathbb{P}\left(\sup_n Y_n = \infty\right)=0 \neq 1 = \mathbb{P}\left(\inf_n Y_n = -\infty\right).$$