1) The restrictive hypotheses on $\psi$ have only been used to establish the implication 1)$\Rightarrow$2).

2) Of course, many other results can be established by the same reasoning, in particular the number field case of Theorem 1.1 of [4]. Let us also recall that the support of the equilibrium measure $\mu_{\varphi}$ is the Julia set $J(\varphi)$. If $J(\varphi)\neq J(\psi)$ at some place, then none of the assertions of Prop. 3.4.3 can possibly hold.

3) The main result of [54] is that a variant of the implication (4)$\Rightarrow$(5) also holds in a more general setting : two semi-positive metrics on a line bundle which define the same measure at a place $v$ differ by multiplication by a constant. The given proof works for curves.

4) We also recall that an implication similar to (1)$\Rightarrow$(5) holds for general metrized line bundles on arithmetic varieties, as proven by [1] : if $\overline{L}$ and $\overline{M}$ are line bundles with adelic metrics such that $h_{\overline{L}}=h_{\overline{M}}$, then $\overline{L}\otimes\overline{M}^{-1}$ is torsion in the Arakelov Picard group $\overline{\operatorname{Pic}}(X)$ : the heights determine the metrics.