Uniformly vanishing functions can retain a nonzero integral
= Uniformly vanishing functions can retain a nonzero integral
{title2=$f_n(x)=n^{-2}x e^{-x/n}$}
On the half-line, $\sup f_n=1/(en)\to0$, but $\int_0^\infty f_n=1$. The change of variable $x=ny$ shows why: the decreasing amplitude is spread over an increasing interval. <Uniform convergence> controls integrals on finite-measure domains; it does not alone control total mass on an infinite-measure domain.