The convergence of martingale: $\sum Y_n- E(Y_n|\mathscr{F}_{n-1})$

Question

I'm reading "Stochastic Processes" by Doob, and I have a question in the following corollary:

The proof is here:

where Theorem 4.1 is:

My question is why $\lim x_n(\omega)$ exists a.e.?

Is it possible that $P(\lim \sup x_n(\omega) = \infty \cap \lim \inf x_n(\omega) = -\infty)>0$?

Answer 1

The series $\sum_{j=1}^{\infty} y_j(\omega)$ converges for almost all $\omega$ for which $\sum_{j=1}^{\infty} p_j(\omega)$ converges, and conversely.

Let $\omega$ such that $\sum_{j=1}^{\infty} y_j(\omega)$ converges, then by the non-negativity

$$x_n(\omega) \leq \sum_{j=1}^n y_j(\omega) \leq \sum_{j=1}^{\infty} y_j(\omega)<\infty,$$

hence $\limsup_{n \to \infty} x_n(\omega)<\infty$. Now it follows from Theorem 4.1 that $\lim_{n \to \infty} x_n(\omega)$ exists and is finite (up to a null set), and this, in turn, implies that $\sum_{j=1}^{\infty} p_j(\omega)<\infty$.

Conversely, if $\sum_{j=1}^{\infty} p_j(\omega)<\infty$ for some $\omega$, then

$$\liminf_{n \to \infty} x_n(\omega) \geq - \sum_{j=1}^{\infty} p_j(\omega)>-\infty$$

and again we find $\lim_{n \to \infty} x_n(\omega)<\infty$ (up to a null set) implying $\sum_{j=1}^{\infty} p_j(\omega)$.

This proves the assertion.