We know that if $\{ f_n \}$ is a sequence of functions bounded by $M$, supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ a.e. $x$, then $f$ is bounded measurable function supported on $E$ for a.e. $x$ and $\int |f_n(x) - f(x)|dx \to 0$ as $n \to 0$.
Here my doubt starts. If $\|f_n\|_{\infty}=\infty$, but if for all every $x\in E$, there exists $M_x>0$ such that $|f_n(x)|\leq M_x$, does bounded convergence theorem still hold?
I would write $M(x)$ instead of $M_x$, because then it's clear that M itself is a function of x (and so also defined on E). If M is integrable the theorem still holds, then it's nothing else then dominated convergence theorem.
If M is not integrable the theorem doesn't hold…