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 $\omega \in \Omega$.
Best Answer
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.