Take $(f_n)$ to be a sequence in $L^1(\mathbb{R})$ and suppose it is true that $(f_n)$ converges in $L^1(R)$ to a function $f \in L^1(\mathbb{R})$. Let $g$ be a function such that $(f_n)$ converges to $g$ (we mean by that $f_n(x) \to g(x)$ for almost all $x \in \mathbb{R}$).
Is there a relationship between $f$ and $g$?
Help is much appreciated
$f=g$ a.e. as $L^1$ convergence implies almost everywhere convergence to $f$ for a subsequence (you can find this as part of the proof of completeness of $L^1$ in any textbook).
Since you know $f_n$ already converges pointwise to $g$, you must have that the same subsequence as above converges to $g$, and hence $f=g$ almost everywhere.