Let $f\in L^1([a,b])$. Does there exist a sequence of continuous functions converging to $f$ in $L^1$ with a function $g\in L^1([a,b])$, where $g$ dominates the sequence a.s., and the set of points where $g$ is discontinuous has measure zero?
Note I asked a similar question here and got an answer: Does there exist a bounding integrable continuous function?
Here the answer shows that for a given sequence, such a $g$ may not exist. However, what I am wondering is if there may exist a sequence and a $g$?(That is for a given sequence such a $g$ may not exist, but there could be another sequence where it does exist?)
EDIT:
I think the proof by @Nate Eldredge here shows that a dominating function may exist: Approximation of a $L^1$ function by a dominated sequence of continuous functions
Now I just need to know if that function can be such that the point of discontinuities have measure zero?