Finite POVM outcome kernel

Let $H$ be a nonzero finite-dimensional complex-inner-product space and fix an ordered basis $\mathcal{B}$ of $H$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Real and complex inner-product spaces and their induced length). Represent each endomorphism by its matrix in $\mathcal{B}$ (Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases), and identify its real and imaginary entries with a point of ${\mathbb{R}}^{2(\dim H{)}^2}$. Give the density set $\mathcal{D}(H)$ (nonnegative operators of trace one, Density operator) the topology whose open sets are intersections of $\mathcal{D}(H)$ with open subsets of that Euclidean coordinate space. Its Borel $\sigma$-algebra is the smallest $\sigma$-algebra containing those open sets; a $\sigma$-algebra is a family of subsets closed under complements and countable unions.

Let $\mathrm{\Omega }$ be the finite outcome set of a POVM $E$ on $H$. A finite probability kernel from $\mathcal{D}(H)$ to $\mathrm{\Omega }$ assigns to each $\rho \in \mathcal{D}(H)$ a finite probability space on $\mathrm{\Omega }$, and requires that $\rho \mapsto K(\rho ,S)$ be Borel measurable for every event $S\subseteq \mathrm{\Omega }$ (Finite probability spaces, outcome weights, events, and event probabilities). The Borel $\sigma$-algebra on $\mathbb{R}$ is generated by its open sets; a real-valued function is Borel measurable when the preimage of every such Borel set belongs to the domain's Borel $\sigma$-algebra.

For the POVM outcome weights from Finite POVMs and their outcome law, define the candidate kernel by

$K_E(\rho ,S):={\sum }_{a\in S}\mathrm{tr}(\rho E_a),S\subseteq \mathrm{\Omega }\mathrm{.}$

Here $\mathrm{tr}$ is the finite-dimensional endomorphism trace (The basis-independent trace of an endomorphism of a finite-dimensional vector space).

That this assignment is a probability kernel and is affine in $\rho$ is proved in The finite POVM law is a probability kernel ↗. The matrix-coordinate $\sigma$-algebra is fixed in the definition; no unmentioned topology on the state set is used.