Leave-one-coordinate rejection partition for a step-up rule

For $1\le r\le m-1$ and $t\in \mathbb{R}$, $\{y_{(r)}\le t\}$ is the finite union over $r$-element index sets $J$ of ${\bigcap }_{j\in J}\{y_j\le t\}$, since at least $r$ remaining values are at most $t$ exactly when the $r$th ordered value is at most $t$. Each coordinate cylinder is measurable in the stated finite product structure by [F1] and [F3], so these sublevel sets are Borel. Since the closed left rays generate $\mathcal{B}(\mathbb{R})$ by [F4], each order statistic is Borel by [F2]. The coordinate-deletion projection from $[0,1{]}^m$ to $[0,1{]}^{m-1}$ is measurable: preimages of measurable rectangles are finite intersections of coordinate cylinders, and the rectangles generate the product sigma-algebra by [F3]. Thus lifting a Borel set in the remaining-coordinate space gives a Borel set in the full product. If $y\le z$ coordinatewise, at least $r$ coordinates of $z$ are at most $z_{(r)}$; their corresponding coordinates of $y$ are also at most $z_{(r)}$, so $y_{(r)}\le z_{(r)}$. Every remaining-coordinate order statistic is therefore coordinatewise nondecreasing.

Fix $k$ and suppose $p_i\le c_k$. Rank $k$ passes exactly when at least $k-1$ other coordinates are at most $c_k$, which is $y_{(k-1)}\le c_k$; for $k=1$ this holds by $y_{(0)}=0$. For each $j>k$, since $p_i\le c_k\le c_j$, rank $j$ passes exactly when at least $j-1$ other coordinates are at most $c_j$, namely $y_{(j-1)}\le c_j$. Hence $R=k$ exactly when the first condition of $C_{i,k}$ holds and every higher rank fails, which is $p_{-i}\in C_{i,k}$. If $k=m$, $i$ is automatically among the first $k$ ranks. If $k<m$ and $i$ is not among the first $k$ ranks, then $p_{(k+1)}\le p_i\le c_k\le c_{k+1}$ would make rank $k+1$ pass. Conversely, if $R=k$ and $i$ is rejected, then $p_i\le p_{(k)}\le c_k$, rank $k$ passes and every higher rank fails, so the conditions defining $C_{i,k}$ hold.

For any remaining vector $y$, replace coordinate $i$ by $0$. Since $c_1>0$, rank one passes. Every rank occupied by a zero-valued coordinate also passes because all critical values are positive, so $R$ includes all zero-valued coordinates and rejects the inserted coordinate $i$. Thus the resulting rejection count is some $r\in \{1,\dots ,m\}$. By step 2.1, $y\in C_{i,r}$; hence the sets cover the space. If $k<\mathrm{\ell }$ and $y\in C_{i,k}$, its condition at $r=\mathrm{\ell }-1$ gives $y_{(\mathrm{\ell }-1)}>c_{\mathrm{\ell }}$, while membership in $C_{i,\mathrm{\ell }}$ would give $y_{(\mathrm{\ell }-1)}\le c_{\mathrm{\ell }}$. Thus the sets are disjoint. They are Borel by step 1.1 and their finite defining inequalities.

If $y\in C_{i,j}$ for some $j\le k<m$, the defining inequalities include $y_{(r)}>c_{r+1}$ for every $r\ge k$, so $D_{i,k}$ is contained in the displayed intersection. Conversely, suppose those inequalities hold and set $p_i=0$. Rank one passes, and by the zero-coordinate argument in step 3.1 the inserted coordinate $i$ is rejected. No rank $\mathrm{\ell }\ge k+1$ can pass: that would require $y_{(\mathrm{\ell }-1)}\le c_{\mathrm{\ell }}$, contrary to the assumed condition at $r=\mathrm{\ell }-1$. Thus the rejection count is some $j\le k$, and step 2.1 puts $y$ in $C_{i,j}$. This proves the formula; when $k=m$, step 3.1 gives $D_{i,m}$ equal to the whole space. The formula is a finite intersection of Borel order-statistic superlevel sets and is coordinatewise increasing by step 1.1.