We know by Vitali Converse that:
let $\mu(E)<\infty$ and {$h_n$} is a sequence of "nonnegative" integrable functions that converges pointwise $a.e.$ on $E$ to $h=0$.
Then $\lim_{n\rightarrow\infty}\int_Eh_n=0$ iff {$h_n$} is uniformly integrable over $E$.
Why it does not hold without the assumption that {$h_n$} is nonnegative?