Recovery of a loop measure from conformal deficits ($A(U)=\nu ({\mathbf{L}}_D\setminus {\mathbf{L}}_U)$)

For simple loops surrounding a marked point, containment in two simply connected domains is containment in their intersection component containing that point. Finite deficits therefore determine all finite intersections of avoidance events by inclusion-exclusion. Restrict to finite windows $H_r={\mathbf{L}}_D\setminus {\mathbf{L}}_{rD}$ and use the sigma-finite uniqueness theorem for measures; as $r\downarrow 0$ these windows exhaust the pointed loop space. Finiteness of the windows is essential to this argument.