§3.3, proof of Theorem 3.1, p. 17 — d = D^µ on single additions and deletions

Let $X$ be a finite set and let $d$ be a semimetric on $\mathcal A(X)$ satisfying Axiom 3. Let $\mu$ be the measure on $2^X$ whose point masses are $\mu\{b\}=d(\succsim_{ab},\succsim^+_{ab})$ for every $a\ne b$ (assumed to be independent of $a$, which Lemma 3.4 guarantees under Axiom 1). Then for every acyclic order $\succsim$ and distinct $a,b\in X$:

1. if $a,b$ are not $\succ$-comparable and $\succsim\oplus(a,b)$ is acyclic, $d(\succsim,\succsim\oplus(a,b))=D^\mu(\succsim,\succsim\oplus(a,b))$;
2. if $a\succ b$, $d(\succsim,\succsim\ominus(a,b))=D^\mu(\succsim,\succsim\ominus(a,b))$.

In words, $d$ and $D^\mu$ have the same value at every pair $(\succsim,\trianglerighteq)$ in which $\trianglerighteq$ is a one-step perturbation of $\succsim$:

This is the base case of the "only if" direction of Theorem 3.1.

Formalization Note $\mu$ is given by point weights w with the hypothesis $w(b)=d(\succsim_{ab},\succsim^+_{ab})$ for all $a\ne b$; this is the page's definition $\mu\{b\}=m_b$, with its well-definedness (Lemma 3.4) taken as part of the hypothesis.