The spatial recentering action is jointly measurable

Fix a Borel set $B\subseteq {\mathbb{R}}^d$. On parameter space ${\mathbb{R}}^d\times {\mathcal{N}}_{\mathrm{l}\mathrm{f}}({\mathbb{R}}^d)$ define the evaluation kernel $K((x,\mu ),C)=\mu (C)$. Its evaluations are measurable by [F1]; the fixed compact exhaustion $[-n,n{]}^d$ has finite $K((x,\mu ),\cdot )$-mass for every $(x,\mu )$ by local finiteness. The function $((x,\mu ),y)\mapsto {\mathbf{1}}_B(y-x)$ is product measurable because subtraction is continuous and $B$ is Borel.

Applying [F2] to this kernel gives measurability of $(x,\mu )⟼\int {\mathbf{1}}_B(y-x)\mu (dy)=\mu (B+x)=({\theta }_x\mu )(B)\mathrm{.}$ Since this holds for every Borel $B$ and evaluations generate the target sigma-algebra [F1], $(x,\mu )\mapsto {\theta }_x\mu$ is measurable.

The argument applies to every Borel evaluation and uses no selection; the AC and Countable Choice tags record only the carried supplier conventions [A1].