Using this result Under what condition can converge in $L^1$ imply converge a.e.?. Can one infer that we can approximate an $L^1$ a.e by a continuous function on a finite measure space?
I.e find $h$ cts s.t $\mid h-g \mid < \epsilon $ a.e
Letting $=f_{n}=g-c_{n}$ where $c_{n}$ is a sequence of continuous approximating to $g$.
Other suggestions which are simplier if this is true would be appreciated!
I might have an alternative solution which is less messy; Since the measure is finite convergece in $L^1$ implies convergnce in measure which implies there is a subseq convering a.e we can pick/find some cts function from this subsequence that are arbitrarly close a.e to out $L^1$ function.
Finite measure was added after Ian's comments!
No, Lusins theorem seams to be as good as it gets.