${\sigma }_k$-Yamabe measure

We found a special divergence structure for the ${\sigma }_k$-Yamabe operator and use it to get a monotonicity formula. We also get an interior $L^{\mathrm{\infty }}$ estimate via its $L^1$ norm for the ${\sigma }_k$-Yamabe operator when $1\le k\le \frac{n}{2}$. Combining these two tools, we prove the weak continuity of the ${\sigma }_k$-Yamabe measure with respect to convergence in measure.