ScalingStacks

Proof. (Sketch) Using Lemma 3.5 and the holomorphic volume form formula (13), the neighbourhood of the toric boundary near $\mathcal{A}_{\lambda,\sigma}^{\infty}$ only contributes $O(s^{-l})$ to the normalised measure, where $l=\dim NC_{\Delta}(\sigma)$. The same lemmas imply (16) by summing over contributions from boundary type regions. In the toric region corresponding to the neighbourhood of $\partial\Delta_{\lambda}^{\vee}$, the convergence of the normalised volume measure follows from Prop. 3.2 and formula (12). ∎