The following is a problem I solved recently from Folland's Real Analysis, in particular Problem(2.1.3). I have read a few solutions online, including on this site, but they are different from mine, some quite complicated, and I would appreciate it if someone could confirm that my simple argument is correct.

In the proof, I use the implicit assumption made by Folland that the $\sigma$-algebra on the range space for a real-valued function is always understood to be the Borel $\sigma$-algbera $\mathcal{B}_{\mathbb{R}}$ unless specified otherwise. I make use of two propositions from Folland's book:

Proposition(2.6): If $f,g : X \rightarrow \mathbb{C}$ are $\mathcal{M}$-measurable, then so are $f + g$ and $fg$.

Proposition(2.7): If $\{f_j\}$ is a sequence of $\overline{\mathbb{R}}$-valued $\mathcal{M}$-measurable functions on $(X, \mathcal{M})$, then the functions$$g_1(x) = \sup_j f_j(x),\quadg_2(x) = \inf_j f_j(x)\\g_3(x) = \lim \sup f_j(x),\quadg_4(x) = \lim \inf f_j(x)$$are all $\mathcal{M}$-measurable. If $f(x) = \lim_{j \rightarrow \infty} f_j(x)$ exists for every $x \in X$, then $f$ is $\mathcal{M}$-measurable.

Problem: Let $(X, \mathcal{M}, \mu)$ be a measure space. If ${f_n}$ is a sequence of $\overline{\mathbb{R}}$-valued $\mathcal{M}$-measurable functions on $X$, then $\{x\ |\ \lim f_n(x) \text{ exists}\}$ is a measurable set.

Proof: Let $f_i = \lim \inf f_n(x)$, $f_s = \lim \sup f_n(x)$, and let $F = \{x\ :\ |f_i(x)| < \infty \text{ or } |f_s(x)| < \infty\}$. Then define $$g(x) = \chi_F(x)(f_i(x) - f_s(x)) + (1 - \chi_F(x))\beta$$ for some choice of $\beta \in \mathbb{R}_{>0}$. We write $\chi_F$ to denote the indicator function on the set $F$, and as usual, we assume the convention that $0 \cdot \pm \infty = 0$. By Proposition(2.7), both $f_i$ and $f_s$ are $\mathcal{M}$-measurable functions. $\mathbb{R}$ is a Borel set, meaning that both $f_i^{-1}(\mathbb{R}), f_s^{-1}(\mathbb{R}) \in \mathcal{M}$, and hence $F = f_i^{-1}(\mathbb{R}) \cup f_s^{-1}(\mathbb{R})$ is measurable. Therefore $\chi_F$ is a simple $\mathcal{M}$-measurable function, and now by repeated application of Proposition(2.6), we conclude that $g$ is $\mathcal{M}$-measurable too.Moreover, notice that our definition of $g$ ensures that it is well defined in the case where both $f_i$ and $f_s$ take infinite values. Finally, observe that$$ \{x\ |\ \lim f_n(x) \text{ exists}\}\ =\ g^{-1}(\{0\})$$since $\lim f_n(x)$ exists if and only if $\lim \inf f_n(x) = \lim \sup f_n(x)$. The set $\{0\}$ is a Borel set, and therefore $g^{-1}(\{0\}) \in \mathcal{M}$, since $g$ is $\mathcal{M}$-measurable. That is the set $\{x\ |\ \lim f_n(x) \text{ exists} \}$ is a measurable set.

Edit(1)

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@Ramiro has raised an interesting point. They claim that $\lim f_n(x)$ exists in the case that $\lim \sup f_n(x) = \lim \inf f_n(x) = \pm \infty$.Furthermore,

the sequence $f_n(x)$ taking values on the extended real line converges if and only if the limit inferior and limit superior are equal (no need to be finite).A sequence taking values on the extended real line may converge to $-\infty$ or $+\infty$.

I have no disagreement about the problem statement, and I certainly understand that we are working on $\overline{\mathbb{R}}$.My (additional) questions are with respect to the difference between when a limit exists and when it converges, which is an essential point to this problem.Up to now, I had presumed that these two expressions are equivalent.A brief inquiry has led me to see that, when working on the extended real line, some people indeed differentiate,and if $\lim x_n = \infty$ for a sequence $x_n \in \overline{\mathbb{R}}$, they will write that the limit exists but that it diverges to $\infty$,in contrast to the limit simply diverging and not existing.However, looking at all my books on analysis by Alhfors, Rudin, Lange, Bogachev, Halmos, ..., I do not find a definition of convergence to $\pm \infty$. Convergence is always defined in terms of a (finite) metric.Similarly, on page 11 of section 0.5, Folland defines convergence of sequences in $\overline{\mathbb{R}}$ to be when the limit inferior and limit superior are both equal and finite.Therefore,

a) Is there a standard textbook that defines $\lim x_n$ on the extended real line to exist in the case that $\lim x_n = \pm \infty$?

b) Is there a standard textbook that defines $\lim x_n$ on the extended real line to converge in the case that $\lim x_n = \pm \infty$? If there is, can you please clarify this definition? More precisely, the definitions that I have learned always define convergence with respect to a metric $d$ on a space $X$, that is a function $d: X \rightarrow [0, \infty)$ satisfying the properties of a metric. Then what is the metric in this case?

Edit(2 - Bounty)

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@Ramiro - Has provided a detailed and thoughtful response, which partially addresses the issues raised in Edit(1). However, I would like to view the perspective of other members in the community, because I disagree with an essential part of the answer, namely that @Ramiro makes it look as if the question of convergence to $\infty$ is a canonical definition, wherease my brief inquiry makes it look as if this really is a matter of convention. Though @Ramiro insists that the mainstream definition of convergence is in terms of a topology, and though I agree that this indeed is a legitimate (if not elegant) definition, in my books on Analysis by Alhfors, Rudin, Lange, ... , convergence is defined for a metric space $X$ with respect to a metric $d$ for that space, moreover the convergence is to a finite value. As I already mentioned, on page 11 of section 0.5, Folland provides his definition of the convergence of sequences $x_n$ taking values in $\overline{\mathbb{R}}$, and he too defines convergence with respect to a metric, in particular when the limit inferior and limit superior are both equal and finite.

The detailed answer of @Ramiro has convinced me that accepting this convention is reasonable from a theoretical point of view, particularly given the metric they have provided, and the Topological perspective on convergence. Furthermore, the references to Royden and Fremlin illustrate that this convention is accepted by numerous mainstream Mathematicians. Nevertheless, the question still remains as to whether this definition is indeed canonical. Although this additional question is in some respects a soft question, the correctness of a solution to Folland's Problem(2.1.3) depends on this point, and therefore it seems worthwhile to examine the matter a bit more carefully.