Circle overlap weights count each arc once

Write $q:\mathbb{R}\to S^1$ for the quotient. The chart domains are $U_1=S^1\setminus \{q(\pi )\}$ and $U_2=S^1\setminus \{q(0)\}$. On one overlap component $x_2=x_1$, and on the other $x_2=x_1+2\pi$. Both derivatives equal one, so the two constant coefficients satisfy the density law.

The disjoint decomposition is $S^1=q((0,\pi ))\bigsqcup q((\pi ,2\pi ))\bigsqcup \{q(0),q(\pi )\}$. The chart formula assigns the two arcs masses $\pi$ and $\pi$, and the endpoints mass zero. Hence ${\mu }_r(S^1)=2\pi$.

For any subordinate pair $({\phi }_1,{\phi }_2)$, translate the negative-angle part of the first chart by $2\pi$. Translation has unit Jacobian. The chart-sum formula becomes ${\int }_0^{2\pi }({\phi }_1(q(t))+{\phi }_2(q(t)))dt={\int }_0^{2\pi }1dt=2\pi$, ignoring only the already null endpoints. Thus overlapping chart weights count each arc once. Empty arcs contribute zero.