= Solution
Since there are finitely many arrival times and each is strictly below one almost surely, $U_*:=\max_iU_i<1$ almost surely. For every $t\geq U_*$ with $t<1$, all weights have arrived and $M_t=0$. Therefore $M_t\to0$ almost surely as $t\uparrow1$.
Nevertheless $\mathbb E M_t=S>0$ for every $t<1$. If the family were <uniformly integrable>, its almost-sure convergence along any deterministic sequence tending to one would imply convergence in $L^1$, and hence convergence of <expectations> to zero. This is impossible. Thus \b[the <martingale> is not <uniformly integrable>]. Its constant expected mass is carried by increasingly rare, late arrivals.
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