Holm rejection events are measurable

By [F1] and [F2], each $p_i$ is measurable into $[0,1]$ with its Borel trace. For indices $a,b$, the difference $p_a-p_b$ is measurable by [F4], so $\{p_a\le p_b\}$ and $\{p_a<p_b\}$ are measurable preimages by [F5] of the Borel half-lines $(-\mathrm{\infty },0]$ and $(-\mathrm{\infty },0)$. For each permutation $\pi$, its ordering cell $C_{\pi }$ is the finite intersection over positions $a<b$ of $\{p_{\pi (a)}\le p_{\pi (b)}\}$ if $\pi (a)<\pi (b)$ and $\{p_{\pi (a)}<p_{\pi (b)}\}$ otherwise. These cells are measurable by [F6], and every $x$ belongs to exactly one because lexicographic ordering with index tie-breaking gives a unique permutation by [F3]. For rank $j$, let $Q_{\pi ,j}=\{p_{\pi (j)}\le \alpha \mathrm{/}(m-j+1)\}$; this pass event and its failure event $Q_{\pi ,j}^c$ are measurable preimages of Borel half-lines by [F1], [F2], and [F5]. For $k\in \{0,\dots ,m\}$, intersect $C_{\pi }$ with the pass events for $j\le k$ and, when $k<m$, the failure event at $k+1$. Empty intersections are omitted when $k=0$ or $k=m$. The resulting measurable cell is exactly the event that the order is $\pi$ and the returned prefix length is $k$, by [F3]; these finitely many cells partition $\mathcal{X}$.

The event $D_i(\alpha )$ is the union of the cells from step 1.1 over all permutations and prefix lengths $k$ with $i\in \{\pi (1),\dots ,\pi (k)\}$. This is a finite union because there are $m!$ permutations and $m+1$ prefix lengths. By [F6], $D_i(\alpha )\in \mathcal{A}$ for every $i$.