The Martingale Convergence Theorem is typically stated that we have convergence to a (finite) random variable $X$ as $X_n \to X$ a.s when the conditions of the theorem are satisfied.
What is meant by a "finite" random variable? Would it be acceptable to replace the phrase "finite random variable" with "bounded random variable"?
In this context, "finite" usually means "almost surely finite", i.e. $|X(\omega)|<\infty$ for almost all $\omega \in \Omega$. Note that this does not imply the boundedness of $X$.