hand, to expose our (very simple) line of argument, and, on the other hand, to deal with our main application to boundary actions ([Theorem \[th:main\]]{}).

\[rem:special\] Condition (iv) from [Theorem \[th:equiv\]]{} is *not*, generally speaking, equivalent just to *existence* of $t>0$ such that the set $\{y\in Gx:
\mu_x(y)\ge t\}$ is finite (i.e., to boundedness of the values of the weights of the measure $\mu_x$). The most manifest example of this is an action with a finite invariant measure, see [Corollary \[cor:Poin\]]{}.

Application to boundary actions
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Hyperbolic spaces and limit sets
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Recall that a non-compact complete proper metric space ${\mathcal X}$ is *Gromov ${\delta}$-hyperbolic* (with ${\delta}\ge 0$) if its metric $d$ satisfies the *${\delta}$-ultrametric inequality* $$(x|z)_o \ge \min\{(x|y)_o,(y|z)_o\} - {\delta}\qquad \forall\,x,y,z,o\in{\mathcal X}\;,$$ where $$(x|y)_o = \frac12 \left[d(o,x) + d(o,y) - d(x,y) \right]$$ is the *Gromov product*. In addition we require that the space $X$ be *separable*. On the other hand, we do *not* require the space ${\mathcal X}$ to be *geodesic*. This class of spaces contains Cartan–Hadamard manifolds with pinched sectional curvatures (in particular, the classical hyperbolic spaces of constant negative curvature) and metric trees, see [@Gromov87; @Ghys-delaHarpe90] for more details.

A Gromov hyperbolic space ${\mathcal X}$ admits a 