In the Dominated Convergence Theorem, we usually assume that $|f_n| \le g$ for some integrable function $g$. However, what is a counter-example where $f_n$ are not dominated by an integrable function but only by their pointwise limit itself, $f$?
That is, $|f_n| \le f$ for all $n$, and $f$ is merely measurable, and the limit of the sequence $(f_n)$ of measurable functions?
EDIT: Or does one not exist?