Fixed-time Aalen–Johansen outputs are well-defined statistics

For every $i,j$, the predicates $U_i\le t$, $U_i\le U_j$, $U_i=U_j$, and each finite-valued cause-status predicate define Borel subsets of $\mathcal{X}$: the coordinate-comparison sets are closed in the Euclidean sample space, and status slices are Borel in the finite status space. Hence $Y(U_i)$ and $\mathrm{\Delta }{N}_k(U_i)$ are finite sums of measurable indicators by [F1] and [F4]. If $i$ is an event at $u=U_i$, then its indicator is included in $Y(u)$ and in $\mathrm{\Delta }N(u)$, so $Y(u)\ge \mathrm{\Delta }N(u)\ge 1$.

For each event record the ratio $\mathrm{\Delta }{N}_k(U_i)\mathrm{/}Y(U_i)$ has finite range, since both counts lie in $\{0,\dots ,n\}$ and the denominator is in $\{1,\dots ,n\}$; each level set is a finite union of Borel count-level sets, so the ratio is measurable. The indicator selecting the first record at each distinct event time and the ordering of the at most $n$ event times are also determined by finitely many Borel equalities and inequalities. The updates in [F1] therefore form finite sums and products of measurable functions; [F4] makes every coordinate of the output vector measurable. By [F2] and [F3], this vector is a statistic, and step 1.1 proves its denominators are positive.