Vitali's convergence theorem - converse

Question

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?

Answer 1

Let $E = [-1,1]$ with Lebesgue measure and take $h_n = n (1_{(0, 1/n)} - 1_{(-1/n, 0)})$. We have $h_n \to 0$ pointwise and $\int h_n = 0$ for every $n$, but $\{h_n\}$ is not uniformly integrable.