I was thinking about the following question regarding convergence almost everywhere (a.e.) and convergence in $L^1$:
Assume we know that there is an $f \in L^1$ and a measurable $g$ such that a sequence $f_n$ converges to $f$ in the $L^1$-norm and at the same time converges to $g$ a.e.. Can we conclude that $f=g$ a.e.?