To show that Egoroff's Theorem continues to hold if the convergence is point-wise a.e. and $f$ is finite a.e. on $E$.
And Egoroff's Theorem states that if $E$ is a set of finite measure and $\{f_n\}$ is a sequence of measurable functions that converges pointwise on $E$ to the real valued function $f$, then there exists a closed set $F$ that is roughly the same size as $E$ on which $\{f_n\}$ converges uniformly.