We are interested in the following class of measures, motivated by Calabi-Yau measures (cf. section 3.1). Let $a_{i}$ be non-negative real numbers assigned to $i\in I$, with $\min a_{i}=0$. Let

$$m=\max\{|J|-1:E_{J}\neq\emptyset,a_{i}=0\text{ for }i\in J\}.$$

We say the measures $d\mu_{t}$ on $X_{t}$ satisfy a uniform upper bound of class $(a_{i})$, if on the local charts of each $E_{J}^{0}$,

$$d\mu_{t}\leq\frac{C}{|\log|t||^{m}}|z_{0}|^{2a_{0}}\cdots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{k}\wedge d\bar{z}_{k}.$$

(4)

The normalisation factor ensures $\int_{X_{t}}d\mu_{t}\leq C$ independent of $t$, by a straightforward local calculation.