Is there a simple example that shows Egorov's theorem fails when the domain $E$ is finite but the limit function $f$ is infinite in a set of positive measure? That is $m(E)<\infty$, $f_k \to f$ a.e. on $E$ with $f_k$ finite but $f$ infinite in a set of positive measure.
Egoroff's Theorem does hold even if $f$ is not finite. By splitting the space into the parts where $f$ is finite and where it is infinity you can concentrate on the case when $f=\infty$ almost eveywhere. Thus, we have $f_n \to \infty$ almost everywhere. Now $\frac {f_n} {1+|f_n|} \to 1$ almost everywhere. Given $\epsilon >0$ we can find $F \subset E$ such that $m(F)<\epsilon$ and $\frac {f_n} {1+|f_n|} \to 1$ uniformly on $E\setminus F$ (by the standard form of the theorem). But this implies $f_n \to \infty$ uniformly on $E\setminus F$.