So I am to prove that, for a measure space $(\Omega,\mathcal{A},\mu)$ and sequences of extended real-valued measurable functions $f_n,g_n$ from $(\Omega,\mathcal{A})$ to the extended real line, such that \begin{equation}\label{1}\tag{1} \lim_{n\to\infty} f_n = f \text{ a.e.}, \quad \lim_{n\to\infty} g_n = g \text{ a.e.}, \quad \forall n \in \mathbb{N}: f_n = g_n \text{ a.e.}, \end{equation}
we have $f=g$ a.e..
As an attempt, I let \begin{equation} A := \{\omega\in\Omega : f(\omega) = g(\omega)\}, \end{equation}
such that for \begin{equation} A_n := \{\omega\in\Omega : g_n(\omega) = f_n(\omega)\}, \end{equation} \begin{equation} B_1 := \{\omega\in\Omega : lim_{n\to\infty} f_n(\omega) = f(\omega)\}, \end{equation} \begin{equation} B_2 := \{\omega\in\Omega : lim_{n\to\infty} g_n(\omega) = g(\omega)\}, \end{equation}
we have \begin{equation} \left(\bigcap_{n=1}^\infty A_n\right) \cap B_1 \cap B_2 \subset A, \end{equation}
i.e.
\begin{equation} A^c \subset \left[\left(\bigcap_{n=1}^\infty A_n\right) \cap B_1 \cap B_2\right]^c = \left(\bigcup_{n=1}^\infty A_n^c\right) \cup B_1^c \cup B_2^c. \end{equation}
Now, by the subadditivity of measures, and by the assumptions \eqref{1}, $\mu(A^c) = 0$ since \begin{equation} 0 \leqslant \mu(A^c) \leqslant \sum_{n=1}^\infty \mu(A_n)^c + \mu(B_1^c) + \mu(B_2^c) = 0, \end{equation}
so it would follow that $f=g$ a.e.. However, since it is not given that $(\Omega,\mathcal{A},\mu)$ is a complete measure space, we do not know if $A^c\in\mathcal{A}$, so $\mu(A^c)$ may be undefined. Which approach would be appropriate? Thanks in advance!
$B_1^c$ and $B_2^c$ and each $A_n^c$ is a null set so $D=B_1^c\cup B_2^c\cup (\cup_nA_n^c)$ is a null set, and $f(x)=g(x)$ for every $x\in D^c.$
It may be that $f(x)=g(x)$ for some $x\in D$ but that is moot. We do not want to show that the complement of $S=\{x:f(x)=g(x)\}$ is null. If the measure is incomplete then $S^c$ may be a non-measurable set. What we need is a null set $D\supset S^c$.