Uniform-arrival residual mass martingale ($M_t={\sum }_i{\theta }_i{1}_{\{U_i>t\}}/(1-t)$)

For finitely many independent uniform arrival times on $[0,1]$, and fixed positive weights, normalize the unarrived mass by the remaining time. Conditional on not having arrived by $s$, each arrival remains uniform on $(s,1)$, giving the martingale identity. The process is càdlàg, nonnegative, and has constant mean equal to the total weight. It vanishes eventually before time one on each typical path, so it is not uniformly integrable when the total weight is positive.

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.