Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space and let $$X_1,X_2,X_3,... \: \Omega \rightarrow \mathbb{R} $$ be a sequence of random variables. Moreover, let there be an event $A \subseteq \Omega$ with $\mathbb{P}(A) = 1$ such that, for all $\omega \in A$, it holds that $$ \lim_{n \rightarrow \infty} X_n(w) $$ exists and is finite.
From all of this, how can I rigorously construct a random variable $$ X: \Omega \rightarrow \mathbb{R}$$ for which it holds that $$ X_n \rightarrow X $$ almost surely, without setting $$ X := X_n \mathbb{1}_{A} \quad? $$ Note that, since $X$ is a random variable, it of course needs to be $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable. Furthermore, note that the range of $X$ should stay finite and should exclude the possibility, that $\lvert X(\omega) \rvert = \infty $ for some $\omega \in \Omega$.
Let $X(\omega)=\lim\sup X_n(\omega)$ if $\lim\sup X_n(\omega) \in \mathbb R$ and $X(\omega)=0$ otherwise. This $X$ has the desired properties.