Let $\{ f_n \}$ be a sequence of measurable functions on $E$ that converges to the real-valued $f$ pointwise on $E$. Show that $E = \bigcup E_k$, where for each index $k$, $E_k$ is measurable, and $\{ f_n \}$ converges uniformly to $f$ on each $E_k$ if $k > 1$, and $m( E_1) = 0$.
my attempt: if $E$ has finite measure then we can apply Egoroff theorem, but how do we get set $E_1$ of measure exactly zero? next when $E$ has infinite measure then case is even more difficult is not it? need help.