The finite POVM law is a probability kernel

Fix $\rho \in \mathcal{D}(H)$. By [F2], the weights $p_{\rho }(a)$ are nonnegative and sum to one, so [F3] gives a finite probability space on $\mathrm{\Omega }$. Its probability for an event $S$ is ${\sum }_{a\in S}{p}_{\rho }(a)$, which equals $K_E(\rho ,S)$ by [F1]. Thus the kernel value at each input is a probability measure on the finite outcome set.

Fix $S\subseteq \mathrm{\Omega }$ and use the ordered basis defining the matrix coordinates. In those coordinates, each summand is $\mathrm{tr}(\rho E_a)={\sum }_{i,j}{\rho }_{ij}(E_a{)}_{ji}$ by [F4] and the row-by-column product formula [F8]; it is a finite linear combination of coordinate projections $\rho \mapsto {\rho }_{ij}$. The projections are continuous in the matrix-coordinate topology, so each summand and the finite sum $g_S(\rho )=K_E(\rho ,S)$ are continuous.

For every open $U\subseteq \mathbb{R}$, continuity in step 1.2 makes $g_S^{-1}(U)$ open and hence Borel. The collection of subsets $B\subseteq \mathbb{R}$ for which $g_S^{-1}(B)$ is Borel is a $\sigma$-algebra, because inverse images preserve complements and countable unions; it contains every open set by the same continuity. It therefore contains the Borel $\sigma$-algebra, proving that $g_S$ is Borel measurable.

Let $\lambda \in [0,1]$ and $\rho ,\sigma \in \mathcal{D}(H)$. Put $T=\lambda \rho +(1-\lambda )\sigma$. Since $\lambda$ and $1-\lambda$ are real and $\rho ,\sigma$ are self-adjoint by [F6]–[F7], for every $x,y\in H$ one has $⟨Tx,y⟩=\lambda ⟨\rho x,y⟩+(1-\lambda )⟨\sigma x,y⟩=⟨x,Ty⟩$. Thus $T$ is self-adjoint; also $⟨Tx,x⟩=\lambda ⟨\rho x,x⟩+(1-\lambda )⟨\sigma x,x⟩\ge 0$, so $T$ is nonnegative by [F7]. It has trace one by [F6] and [F5], so it lies in $\mathcal{D}(H)$. Applying [F5] to the finite sum in [F1] gives $K_E(\lambda \rho +(1-\lambda )\sigma ,S)=\lambda K_E(\rho ,S)+(1-\lambda )K_E(\sigma ,S)$. This proves affinity.