Finite-grid risk control by monotone envelopes

For each $i$, ${\tilde{L}}_i({\lambda }_j)$ is a maximum of finitely many measurable coordinates, so it is measurable and lies in $[0,B]$. The set of indices in the maximum shrinks as $j$ increases, hence ${\tilde{L}}_i({\lambda }_{j+1})\le {\tilde{L}}_i({\lambda }_j)$. At the endpoint, ${\tilde{L}}_i({\lambda }_M)=L_i({\lambda }_M)\le \alpha$. The same deterministic row map is applied to every $W_i$, and it commutes with row permutations; therefore the transformed rows remain exchangeable. Each calibration threshold is a finite sum of transformed coordinates, and each first-pass or fallback event is a finite Boolean combination of Borel threshold sets. Thus the selected parameter $\tilde{\lambda }(W)$ and the losses $g(W)=L_{n+1}(\tilde{\lambda }(W))$ and $h(W)={\tilde{L}}_{n+1}(\tilde{\lambda }(W))$ are Borel functions of the full matrix.

By [F2], applying the calibration rule to the transformed rows gives $\mathbb{E}[{\tilde{L}}_{n+1}(\tilde{\lambda })]\le \alpha$.

For every grid index $j$, the maximum defining ${\tilde{L}}_{n+1}({\lambda }_j)$ includes $L_{n+1}({\lambda }_j)$, so $L_{n+1}(\tilde{\lambda })\le {\tilde{L}}_{n+1}(\tilde{\lambda })$ almost surely. Both variables are bounded and integrable; [F3] and step 1.2 give $\mathbb{E}[L_{n+1}(\tilde{\lambda })]\le \alpha$.

For the conditional clause, let $Q_{\omega }$ be the stated regular conditional distribution of the full matrix $W=(W_1,\dots ,W_{n+1})$ given $\mathcal{T}$. For almost every $\omega$, its row law is exchangeable, its coordinates lie in $[0,B]$, and its endpoint losses are at most $\alpha$ almost surely. Applying the same Borel tail-maximum map to each row preserves exchangeability under $Q_{\omega }$, gives nonincreasing bounded rows, and leaves the endpoint unchanged. The marginal assertion of [F2], applied under each such probability measure $Q_{\omega }$ to the transformed rows, gives $\int hd{Q}_{\omega }\le \alpha$. Since $0\le g\le h\le B$ pointwise, [F3] under $Q_{\omega }$ yields $\int gd{Q}_{\omega }\le \alpha$.

For each $m\ge 1$, put $g_m=2^{-m}\lfloor 2^{m}g\rfloor$. These are finite-valued Borel simple functions increasing pointwise to $g$. Write $g_m={\sum }_r{c}_{m,r}{\mathbf{1}}_{A_{m,r}}$ for Borel level sets $A_{m,r}$ in the full matrix space. By the kernel identity [F4], for every $H\in \mathcal{T}$, ${\int }_H\int g_{m}d{Q}_{\omega }d\mathbb{P}(\omega )={\int }_H{g}_m(W)d\mathbb{P}\mathrm{.}$ The kernel integrals of $g_m$ are $\mathcal{T}$-measurable finite sums of kernel evaluations. By [F6], they increase to $k(\omega )=\int gd{Q}_{\omega }$, and the right-hand integrals increase to ${\int }_{H}g(W)d\mathbb{P}$. Applying [F6] on both sides gives the same identity with $g$ in place of $g_m$. Thus $k$ is $\mathcal{T}$-measurable, bounded, and has the defining integral identities of a conditional-expectation version by [F5]. Step 2.2 gives $k\le \alpha$ almost surely, proving the conditional bound.