Counts Generate the Sigma-Algebra of Finite Configurations

On the component of configurations with exactly $n$ points, $N_B(x_1,\dots ,x_n)={\sum }_{j=1}^n{\mathbf{1}}_B(x_j)$ is Borel for every Borel $B\subseteq C$. The count of $C$ identifies the component $n$, so every count map is Borel on the full disjoint union. Thus the count-generated sigma-algebra is contained in the sorted-list Borel sigma-algebra.

Each $I_{m,k}$ is Borel as the intersection of a closed ray and an open ray. Distinct interval interiors are disjoint and an adjacent endpoint belongs only to the interval beginning there. Induction on the number of adjacent intervals gives ${\bigcup }_{k<j}{I}_{m,k}=[0,j{2}^{-m})$; at $j=2^m$ this is $[0,1)$. Thus they partition the reference interval. Each interval is the union of its two intervals at the next level, so the partition refines and its mesh tends to zero. Cartesian products give the corresponding Borel partitions of $C$.

Let $(T,\mathcal{T})$ be any measurable space. For $H\in \mathcal{B}(C)\otimes \mathcal{T}$, define $M_H(\eta ,t)={\sum }_{x\in \eta }{\mathbf{1}}_H(x,t)$. The class of such $H$ for which $M_H$ is measurable with respect to the count sigma-algebra times $\mathcal{T}$ contains every rectangle $A\times D$, since $M_{A\times D}(\eta ,t)=N_A(\eta ){\mathbf{1}}_D(t)$. It is closed under relative complements because $M_{(C\times T)\setminus H}=N_C-M_H$, and under disjoint countable unions by pointwise convergence of finite partial sums. Rectangles form a generating pi-system, so [F2] proves measurability for every product-measurable $H$.

Fix a component with $n$ distinct points and an index $k\le n$. Recover the coordinates of the $k$th lexicographic point recursively. Initially the selected prefix is empty and its rank is $r=k$. At coordinate $j$, with measurable prefix $p=(a_1,\dots ,a_{j-1})$, use step 2.1 on the Borel set $H_j=\{(x,p,t):x_{\mathrm{\ell }}=p_{\mathrm{\ell }}(\mathrm{\ell }<j),x_j<t\}$, where $t\in [0,1]$. For each dyadic interval $I=[u,v)$ in coordinate $j$, the number of points matching the prefix and having $x_j\in I$ is $M_{H_j}(\eta ,p,v)-M_{H_j}(\eta ,p,u)$, hence is measurable. The interval containing the $r$th such coordinate is the first whose cumulative count reaches $r$; the nested interval closures have diameters tending to zero, so their left endpoints converge to that coordinate $a_j$. The count of matching points with $x_j<a_j$ is $M_{H_j}(\eta ,p,a_j)$, measurable by step 2.1 after composition with the measurable prefix and $a_j$. Subtracting it from $r$ gives the positive rank among points with the extended prefix.

The recursive choice in step 3.1 is the usual lexicographic ordering rule: among points with a fixed prefix, sort by the next coordinate, retain the value at the current rank, then subtract the points strictly below it to obtain the rank among ties. After coordinate $d$, distinctness leaves exactly one point, namely $x_k$. Thus all coordinate functions of every sorted point are count-measurable. The Borel sigma-algebra on the sorted-list component is generated by its coordinate maps, so it is contained in the count-generated sigma-algebra there.

The count $N_C$ partitions the full configuration space into its countable components. Applying step 4.1 on each component shows that every sorted-list Borel set is count-measurable; step 1.1 proves the reverse inclusion. The displayed sigma-algebra equality follows.