For a probability distribution function $F$ supported on a bounded interval $[a,b]$, the integrated survival function (ISF) is defined as $$\Psi_F(t)=\mathbb E_F\max\{X-t,0\}=\int_t^b(1-F(x))d x.$$ Clearly, if two distributions have the same ISF then they are equal. The question is whether uniform convergence in ISF implies uniform convergence in distribution function, or at least just point-wise.
Consider distribution functions $F,F_1,F_2,\dots$ all supported on a bounded interval $[a,b]$. Suppose $\sup_t\left|\Psi_{F_n}(t)-\Psi_F(t)\right|\to0$. Is it true that $\sup_t\left|{F_n}(t)-F(t)\right|\to0$, or at least that $\forall t\ \left|{F_n}(t)-F(t)\right|\to0$? Does it help to assume that $\forall n,t\ \Psi_{F_n}(t)\geq\Psi_F(t)$?
For a negative result, assume that $F$ and $G$ are the CDFs of the Dirac measures at $x$ and $y$. Then $\|\Psi_F-\Psi_G\|_\infty=|x-y|$ while, if $x\ne y$, there exists some $t$ such that $|F(t)-G(t)|=1$, hence the former can be small even when the latter is large. Normal distributions with means $x$ and $y$ and small variances show that the same phenomenon can occur with smooth CDFs.