is closed under a number of operations. Clearly it is symmetric: $(A,B)$ is a critical pair if and only if $(B,A)$ is. It is also obvious that if $(A,B)$ is a critical pair, then so is $(A+x,B+y)$ for any $x,y \in G$, where $A+x \coloneqq \{ a +x: a \in A \}$ denotes the translate of $A$ by $x$. Next, we observe that it is insensitive to asymptotic equivalence:

\[crit-equiv\] Suppose that $(A,B)$ is a critical pair, and that $A'$ is asymptotically equivalent to $A$. Then $(A',B)$ is also a critical pair.

Of course by symmetry, the same statement holds if we replace $B$ by an asymptotically equivalent $B'$. Thus one only needs to know $A,B$ up to asymptotic equivalence to determine if $(A,B)$ form a critical pair.

By hypothesis, there exists an infinitesimal ${\varepsilon}>0$ such that $$\mu_G(A' \Delta A) \leq {\varepsilon},$$ which implies the pointwise bound $$| 1_{A'} * 1_B - 1_A * 1_B | \leq {\varepsilon}$$ and hence we have the inclusion $$A' +_{\delta+{\varepsilon}} B \subset A +_\delta B$$ for any $\delta>0$. On the other hand, as $(A,B)$ is a critical pair, there exists an infinitesimal $\delta>0$ such that $$\mu_G(A +_\delta B) \leq \mu_G(A) + \mu_G(B) + o(1),$$ 