In the situation of Proposition 4.13, the limit depends only on the metrics $\|\cdot\|_{i}$ but not on the sequences $(\|\cdot\|_{i,k})_{k\in\mathbb{N}}$. Namely, if $(\|\cdot\|^{\prime}_{i,k})_{k\in\mathbb{N}}$ are other sequences converging uniformly to $\|\cdot\|_{i}$ then the sequences $(\|\cdot\|^{\prime\prime}_{i,k})_{k\in\mathbb{N}}$ defined by

$$\|\cdot\|^{\prime\prime}_{i,k}:=\begin{cases}\|\cdot\|_{i,\frac{k}{2}},\;k\text{ even}\\ \|\cdot\|^{\prime}_{i,\frac{k-1}{2}},\;k\text{ odd}\end{cases}$$

converge uniformly to $\|\cdot\|_{i}$. As $(\|\cdot\|_{i,k})_{k\in\mathbb{N}}$ and $(\|\cdot\|^{\prime}_{i,k})_{k\in\mathbb{N}}$ are subsequences of $\|\cdot\|^{\prime\prime}_{i,k}$ the limit of the measures is the same. We denote the measure corresponding to the metrics $\|\cdot\|_{i}$ by $c_{1}(\overline{L}_{1})\wedge...\wedge c_{1}(\overline{L}_{n})$.