of $L^2$ functions, is a continuous function of $x$ that equals $\mu_G(A)$ when $x=0$, and has a mean value of $\mu_G(A)^2$ on $G$ by Fubini’s theorem. The claim then follows from the intermediate value theorem and the connectedness of $G$.

By Lemma \[cont\], there exists $x \in G$ such that $\mu_G( A \cap (x+A) ) = a - {\varepsilon}$, and hence by inclusion-exclusion $\mu_G(A \cup (x+A) ) = a+{\varepsilon}$. From with $A_1,A_2$ replaced by $A, x+A$ we have $$c( A) + c(x+A) \geq c(A \cap (x+A)) + c(A \cup (x+A)) \geq f(a-{\varepsilon}) + f(a+{\varepsilon}).$$ By translation invariance we have $c(x+A) = c(A)$, hence $$2c(A) \geq f(a-{\varepsilon}) + f(a+{\varepsilon}).$$ Taking infima over all $A$ with $\mu_G(A) = a$, we contradict , and the claim follows.

With some minor notational modifications, this argument also works for nonabelian compact connected groups; see [@blog].

Proof of Corollary \[cor\] {#cor-sec}
==========================

We now prove Corollary \[cor\]. Suppose that $A,B$ are compact subsets of a compact connected abelian group $G$ are such that equality holds in . We may assume that $\mu_G(A), \mu_G(B), 1 - \mu_G(A)-\mu_G(B) > 0$, since we are done otherwise. Applying Theorem \[inv-1\], we conclude that there exist sequences $\phi_n^{-1}(I_n), \phi_n^{-1}(J_n)$ of parallel 