Theorem 8.12.2655B0BBF4F9C

Proof. Symmetry and 8.12.1, applied to the $r$ slots, give $\mu _ r(w)=\mu _ r(v)$ on $\{ u<v\} $ and $\mu _ r(w)=\mu _ r(u)$ on $\{ u>v\} $. To include the contact set, put $w_\epsilon =\max \{ u+\epsilon ,v\} $. Strict locality and invariance under addition of constants give

The invariance follows from the smooth products and their joint decreasing limits. As $\epsilon \downarrow 0$, $w_\epsilon \downarrow w$, so the left side converges weakly to $\mu _ r(w)$. Against any nonnegative continuous compactly supported test function, the right side converges by dominated convergence for the fixed measure $\mu _ r(u)$ to $\mathbf1_{\{ u\geq v\} }\mu _ r(u)$. This proves the measure inequality on the contact set as well as on $\{ u>v\} $. On its disjoint complement $\{ u<v\} $, the already proved equality supplies $\mu _ r(v)$. Restricting the positive measures to these two Borel sets and adding gives the assertion. Exchanging $u,v$ gives its other form. $\square$