Vtyjkyuk

We know that if $(h_n)$ is a sequence of nonnegative integrable functions on a measurable subset $E$ of $mathbb R$ with finite measure such that $h_nto 0$ pointwise a.e., then $limlimits_nto inftyintlimits_E h_n dm=0$ iff $(h_n)$ is uniformly integrable.

I want to have an example of non-uniformly integrable $(h_n)$, where $h_n$'s are not nonnegative and $h_nto 0$, $limlimits_nto inftyintlimits_E h_n dm=0$. This question is from Royden's Real Analysis. Any hint will be appreciated.

Consider $E=(-3,3)$, and for $n=1,2, ldots$ define the functions
beginalignh_n(x) :=
begincases
-2n - n^2x, ; text if x in left[-frac2n, ,-frac1n right] , \
n^2x , ; text if x in left[-frac1n , , ,frac1nright] , \
2n - n^2x , ; text if x in left[frac1n , ,frac2nright], \
0, ; text if x in left(-3, -frac2nright) cup left(frac2n, 3 right). endcases
endalign

Notice $limlimits_nto infty h_n(x)=0$ and $limlimits_nto inftyintlimits_E h_n dm=0$. Let $delta>0$ be given. By the Archimedean Property, we may find a positive integer $N$ such that $N delta > 4$. So we have that $mleft( left[-frac2N, frac2N right]right)< delta$ but notice $int_-frac2N^frac2N left| h_n right| dm=2$ whenever $n geq N$*. Hence the sequence of functions $h_n_n=1^infty$ is not uniformly integrable.

*Note: Technically it suffices to observe that $int_-frac2N^frac2N left| h_N right| dm=2$ since
$neg left(forall varepsilonleft(varepsilon>0 implies exists delta
left( delta>0 land forall n forall A left(m(A)<delta implies int_A |,h_n| dm < varepsilon right)right) right) right)$

$iff exists varepsilon left( varepsilon>0 land forall delta left( delta>0 implies exists N exists A left(m(A)<delta land int_A |,h_N| dm geq varepsilonright)right) right)$.