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For the uniform-arrival residual mass martingale with total weight $0<S<1$, the first crossing of level one happens continuously: its only jumps are downward, and an arrival almost surely misses the finitely many deterministic possible crossing times. Stop at that crossing or at $r<1$. The stopped values lie between zero and one, and their limit as $r\uparrow 1$ is the crossing indicator. Optional stopping and dominated convergence identify its probability as $S$. At $S=1$ the strict boundary inequality already fails at time zero.

Write the residual process asFor $s<t<1$, the filtration tells us whether each $U_i$ has arrived by time $s$, and its exact arrival time if it has. On $\{U_i>s\}$, its conditional law is uniform on $(s,1)$. Independence of the original $U_i$ means that the histories of the other arrivals do not alter this law. HenceThis conditional identity can also be checked against products of bounded functions of the censored arrival times and extended to ${\mathcal{F}}_s$ by the Monotone class theorem. Summing the identity gives $\mathbb{E}[M_t\mid {\mathcal{F}}_s]=M_s$.

Each $M_t$ is adapted and integrable, withIts paths are smooth between the finitely many arrival times and have downward jumps of size ${\theta }_i/(1-U_i)$ at arrivals before time one. The indicators use $U_i>t$, so the value at a jump is the post-jump value: paths are right continuous with finite left limits on $[0,1)$. Thus $M$ is a càdlàg martingale, specifically the uniform-arrival residual mass martingale.