Totality and measurability of the repaired multivariate trimmed-mean selector

Fix the data tuple $x$ and put $M(x)={\max }_i\{\mathrm{\parallel }{\tilde{X}}_i{\mathrm{\parallel }}_2,\mathrm{\parallel }{\tilde{Y}}_i{\mathrm{\parallel }}_2\}$. Every projected observation and each order-statistic endpoint lies in $[-M(x),M(x)]$, so $\mathrm{\mid }{U}_Q(v;x)\mathrm{\mid }\le M(x)+Q$. Hence $z\in \mathrm{\Gamma }(Q)$ implies $\mathrm{\mid }⟨z,v⟩\mathrm{\mid }\le M(x)+(1+2\epsilon )Q$ for every unit $v$ and $\mathrm{\parallel }z{\mathrm{\parallel }}_2\le M(x)+(1+2\epsilon )Q$ by [F4]. Each order statistic of a finite list $a_1,\dots ,a_N$ equals ${\min }_{\mathrm{\mid }J\mathrm{\mid }=k}{\max }_{j\in J}{a}_j$ at rank $k$: every such maximum is at least the $k$th smallest entry, and the first $k$ indices attain it. Thus the directional order statistics, clipping endpoints, and $U_Q(v;x)$ are continuous in $(x,v)$ for fixed $Q$. Each $\mathrm{\Gamma }(v,Q)$ is a closed convex slab; intersections preserve these properties. Therefore $\mathrm{\Gamma }(Q)$ and every $K_j$ are closed and convex; a nonempty $K_j$ is bounded by the displayed radius at $Q=2^j$, hence compact by [F2].

Since $K_j\subseteq K_{j+1}$, $S=\{j:K_j\mathrm{\ne }\mathrm{\varnothing }\}$ is upward closed. If $S$ is nonempty and proper, choose $j_0\in S$ and $j_1\mathrm{\notin }S$; upward closure forces $j_1<j_0$, and the least point of the nonempty finite set $S\cap \{j_1+1,\dots ,j_0\}$ is least in $S$. If $S=\mathbb{Z}$, the nested nonempty compact sets $K_0\supseteq K_{-1}\supseteq K_{-2}\supseteq \cdots$ have the finite-intersection property inside compact $K_0$, so [F5] gives a nonempty compact intersection. This intersection is exactly ${\bigcap }_{i\in \mathbb{Z}}\mathrm{\Gamma }(2^i)$, since the constraints in $K_{-n}$ range over all $i\ge -n$. Each nonempty branch set (a least-index $K_j$ or the all-scale intersection) is compact and convex, so its norm attains a minimum by [F3]. If distinct $z,w$ both minimize squared norm, then $\mathrm{\parallel }(z+w)\mathrm{/}2{\mathrm{\parallel }}_2^2=(\mathrm{\parallel }z{\mathrm{\parallel }}_2^2+\mathrm{\parallel }w{\mathrm{\parallel }}_2^2)\mathrm{/}2-\mathrm{\parallel }z-w{\mathrm{\parallel }}_2^2\mathrm{/}4$ is smaller, a contradiction; the minimum-norm point is unique. The empty-$S$ fallback is the defined first-half mean. Thus every branch is total and single-valued.

For fixed $Q>0$, define $g_Q(x,z)={\max }_{v\in S^{d-1}}\{\mathrm{\mid }⟨z,v⟩-U_Q(v;x)\mathrm{\mid }-2\epsilon Q\}$. It is continuous: the maximand is jointly continuous by step 1.1, the sphere is compact by [F2], and on a compact neighborhood of any $(x,z)$ times the sphere [F8] gives uniform continuity, which bounds the change in the maximum by the supremum change of the maximand. Also, $\mathrm{\Gamma }(Q)=\{z:g_Q(x,z)\le 0\}$, and $g_Q$ is convex in $z$ as a maximum of absolute affine functions minus constants. Put $g_I(x,z)={\sup }_{i\in I}{g}_{2^i}(x,z)$ for $I=I_j=\{i\in \mathbb{Z}:i\ge j\}$ or $I=\mathbb{Z}$. If $Q\ge 2M(x)$, no first-half projection is clipped, so $U_Q(v;x)=⟨\bar{X},v⟩$ and $g_Q(x,z)=\mathrm{\parallel }z-\bar{X}{\mathrm{\parallel }}_2-2\epsilon Q$, where $\bar{X}=N^{-1}{\sum }_i{\tilde{X}}_i$. On any compact neighborhood of $(x,z)$ this identity makes the high-scale terms tend uniformly to $-\mathrm{\infty }$; for $I_j$ they are eventually below $g_{2^j}$, and for $I=\mathbb{Z}$ eventually below $g_1$. Thus the high-scale supremum is locally a finite maximum. For the all-integer index set, define $g_0$ using clipping to $[{\alpha }_v,{\beta }_v]$ and no error allowance; the same compact-sphere maximum argument shows $g_0$ is continuous. Expanding both clipping endpoints by $Q$ changes each clipped observation by at most $Q$, so $\mathrm{\mid }{g}_Q-g_0\mathrm{\mid }\le (1+2\epsilon )Q$ uniformly in $(x,z)$. On a compact neighborhood, all scales $i<-m$ therefore have supremum between $g_0$ and $g_0+(1+2\epsilon )2^{-m}$, while for all sufficiently large $m$ the scales $i>m$ are below $g_1$. Hence $g_I$ is a local uniform limit of the continuous finite maxima $\max \{g_0,g_{2^i}:-m\le i\le m\}$ and is continuous. Being a supremum of convex functions, $g_I$ is convex in $z$.

For $I=I_j$, every $z\in K_j$ satisfies $\mathrm{\parallel }z{\mathrm{\parallel }}_2\le R_j(x)=M(x)+(1+2\epsilon )2^j$ by step 1.1; for $I=\mathbb{Z}$, every feasible $z$ lies in $\mathrm{\Gamma }(1)$ and has norm at most $R_{\mathrm{\infty }}(x)=M(x)+1+2\epsilon$. These radii are positive and continuous. Since $K_I(x)=\{z:g_I(x,z)\le 0\}$, its nonemptiness is equivalent to $m_I(x)\le 0$, where $m_I(x)={\min }_{\mathrm{\parallel }w{\mathrm{\parallel }}_2\le 1}{g}_I(x,R_I(x)w)$. The minimum exists by [F2], [F3]. It is continuous in $x$: on a compact neighborhood of any $x$, [F8] makes $(x,w)\mapsto g_I(x,R_I(x)w)$ uniformly continuous in $x$ uniformly over the compact unit ball, and the difference of two minima is bounded by the supremum of the corresponding function differences. Thus each event $\{x:K_I(x)\mathrm{\ne }\mathrm{\varnothing }\}=\{x:m_I(x)\le 0\}$ is Borel.

Fix one such branch index set $I$ and write $R=R_I(x)$. By [F6], enumerate a countable dense rational subset $(r_a)$ of ${\mathbb{Q}}^d$ and set $d_a=r_a\mathrm{/}\max \{1,\mathrm{\parallel }{r}_a{\mathrm{\parallel }}_2\}$. The radial map is continuous and is the identity on the unit ball, so $(d_a)$ is dense in that ball. Let $h_I(x,w)=g_I(x,Rw)$ and, for each integer $\lambda \ge 1$, minimize $J_{\lambda ,I}(x,w)=R^2\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+\lambda \max \{h_I(x,w),0\}$ over $\mathrm{\parallel }w{\mathrm{\parallel }}_2\le 1$. The function is jointly continuous and strictly convex in $w$: $h_I$ is convex by step 2.2, its positive part is convex, and $R>0$ makes the quadratic strictly convex. Compactness and [F3] give a unique minimizer $w_{\lambda ,I}(x)$. The value $b_{\lambda ,I}(x)={\inf }_a{J}_{\lambda ,I}(x,d_a)$ is Borel as a countable infimum of continuous functions and equals the minimum by density and continuity. For each integer $s\ge 1$, put $A_{a,s}=\{x:J_{\lambda ,I}(x,d_a)<b_{\lambda ,I}(x)+1\mathrm{/}s\}$. The rational points in [F6] give $A_{a,s}={\bigcup }_{q\in \mathbb{Q}}\{x:J_{\lambda ,I}(x,d_a)<q\}\cap \{x:b_{\lambda ,I}(x)>q-1\mathrm{/}s\}$, so $A_{a,s}$ is Borel. Such indices exist by density; the least one's level set is $A_{a,s}\setminus {\bigcup }_{b<a}{A}_{b,s}$, and $x\mapsto d_{a(x,s)}$ is Borel because each open-set preimage is the union of the Borel level sets whose values lie in that open set. Strong convexity gives $J_{\lambda ,I}(x,y)-J_{\lambda ,I}(x,w_{\lambda ,I})\ge R^2\mathrm{\parallel }y-w_{\lambda ,I}{\mathrm{\parallel }}_2^2$: apply the quadratic identity $\mathrm{\parallel }(1-t)w+ty{\mathrm{\parallel }}_2^2=(1-t)\mathrm{\parallel }w{\mathrm{\parallel }}_2^2+t\mathrm{\parallel }y{\mathrm{\parallel }}_2^2-t(1-t)\mathrm{\parallel }y-w{\mathrm{\parallel }}_2^2$, convexity of the penalty, minimality, and let $t\downarrow 0$. Hence $\mathrm{\parallel }{d}_{a(x,s)}-w_{\lambda ,I}(x){\mathrm{\parallel }}_2<1\mathrm{/}(R\sqrt{s})$, so these Borel maps converge pointwise to $w_{\lambda ,I}$. Each selected map is countably valued with Borel level sets, so its coordinate functions are Borel; [F7] makes their pointwise limits measurable. Rational open boxes form a countable basis by [F6], so $w_{\lambda ,I}$ is Borel. For each coordinate $j$, the pair $(R_I(x),w_{\lambda ,I,j}(x))$ is Borel because inverse images of rational rectangles are intersections of coordinate preimages and those rectangles form a countable basis of ${\mathbb{R}}^2$. Multiplication is continuous, so its composition with this pair map is Borel by the inverse-image definition; hence $z_{\lambda ,I}(x)=R_I(x)w_{\lambda ,I}(x)$ is Borel.

On the Borel event $E_I=\{x:K_I(x)\mathrm{\ne }\mathrm{\varnothing }\}$, let $z_{\ast }(x)$ be its unique minimum-norm point from step 2.1 and put $w_{\ast }=z_{\ast }\mathrm{/}R$. Comparing the penalized minimum with the feasible $w_{\ast }$ gives $\mathrm{\parallel }{z}_{\lambda ,I}(x){\mathrm{\parallel }}_2\le \mathrm{\parallel }{z}_{\ast }(x){\mathrm{\parallel }}_2$ and $\max \{g_I(x,z_{\lambda ,I}(x)),0\}\le \mathrm{\parallel }{z}_{\ast }(x){\mathrm{\parallel }}_2^2\mathrm{/}\lambda$. The sequence $z_{\lambda ,I}(x)$ lies in the compact closed ball of radius $R(x)$ by [F2]. Every sequence in a compact metric space has a convergent subsequence: if no point had infinitely many terms in every neighborhood, neighborhoods containing only finitely many term indices would cover the ball, contradicting a finite subcover; if one point did have infinitely many terms in every neighborhood, choosing the least successive indices in shrinking balls would give a convergent subsequence. Thus every subsequence has a convergent subsubsequence. By continuity of $g_I$, each cluster point is feasible; the norm bound makes its norm no greater than $\mathrm{\parallel }{z}_{\ast }(x){\mathrm{\parallel }}_2$, so uniqueness in step 2.1 forces every cluster point to equal $z_{\ast }(x)$. The full sequence therefore converges to $z_{\ast }(x)$: otherwise a subsequence staying a fixed positive distance away would have a different cluster point. The maps equal to $z_{\lambda ,I}$ on $E_I$ and $0$ off it are Borel: for an open set their inverse image is the branch-event intersection with the Borel inverse image under $z_{\lambda ,I}$, with $E_I^c$ added exactly when $0$ belongs to the open set. They converge everywhere, to $z_{\ast }$ on $E_I$ and to $0$ off it; their pointwise limit is Borel by [F7].

The events $S=\mathrm{\varnothing }$, $S=\mathbb{Z}$, and $\min S=j$ are respectively countable intersections or combinations of the Borel nonemptiness events in step 3.1, so they are Borel by [F7]. On each such event the algorithm uses the corresponding Borel branch selector from step 5.1 or the continuous first-half mean. For every open target set, its inverse image under the piecewise map is the countable union of each branch event intersected with that branch map's Borel inverse image; it is Borel. Step 2.1 proves all branches defined and single-valued, so the complete output map is total and Borel.