§3.3, proof of Theorem 3.1, pp. 17–18 — d = D^µ on acyclic orders with equal symmetric parts

Let $X$ be a finite set and let $d$ be a semimetric on $\mathcal A(X)$ satisfying Axioms 1 and 3. Let $\mu$ be the measure on $2^X$ with point masses $\mu\{b\}=d(\succsim_{ab},\succsim^+_{ab})$ for every $a\ne b$. Then for all acyclic orders $\succsim,\trianglerighteq$ with the same symmetric parts,

Together with Axiom 4, which lets one replace each order by its strict part plus the diagonal, this yields the "only if" direction of Theorem 3.1.

Formalization Note As in the previous milestone, $\mu$ is given by point weights w with $w(b)=d(\succsim_{ab},\succsim^+_{ab})$ for all $a\ne b$. The case $\succsim=\trianglerighteq$ is included (both sides are $0$).