Let $G$ be a countable group and $\mathcal{B}$ be the Borel sigma-algebra of $G$. Suppose that $B \in \mathcal{B}$ and that $A$ is a finite subset of $G$. I want to prove that for any fixed configuration $w \in G^A$, $\mathbb{1}_{\{wx_{A^c} \in B\}}(x)$ is $\mathcal{B}_{A^c}$-measurable.
I am out of ideas to prove this. On one hand, it seems obvious, since (sorry for the simplification!) we only consider the coordinates of $x$ outside $A$. On the other hand, I don't know how to prove. The thing is that $B$ might not be $B_{A^c}$-measurable...