Just having a little problem solving this, however it probably is pretty easy and I am just being dumb.
Suppose you have a lebesgue integrable function $f$. The goal is, for any $ \epsilon > 0 $, to find a set $C$ with $ \mu (C) < \infty$ such that $\int_{C^c} |f| d \mu < \epsilon$.
Any ideas on how to construct this set? I think it has to do with the Dominated Convergence Theorem, but I don't see it.