Uniformly vanishing functions can retain a nonzero integral ($f_n(x)=n^{-2}x{e}^{-x/n}$)

On the half-line, $\sup f_n=1/(en)\to 0$, 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.