simple consequence of this fact is the description of the conservative part of a free action $G{\circlearrowleft}(X,m)$ as the set of points $x\in X$ with the property that $$\tag{$*$}
\textsf{there exists $t=t(x)$ such that $\{g\in G:dgm/dm(x)\ge t\}$ is
infinite}$$ ([Theorem \[th:equiv\]]{}(iv)).

The latter result completely trivializes the problem of a geometric description of the conservative part of the boundary action of a discrete group of isometries $G$ of a Gromov hyperbolic space ${\mathcal X}$ with respect to a certain natural measure class, which is our main application.

More precisely, for any boundary point ${\omega}\in{\partial}{\mathcal X}$ and any $x,y\in X$ let ${\beta}_{\omega}(x,y) = \limsup_z [d(y,z)-d(x,z)]$, where $z\in{\mathcal X}$ converges to ${\omega}$ in the hyperbolic compactification. For CAT($-1$) spaces ${\beta}_{\omega}$ are the usual *Busemann cocycles*, whereas in the general case the cocycle identity is satisfied up to a uniformly bounded additive error only, so that we have to call them *Busemann quasi-cocycles*. Then one can look for a family ${\lambda}=\{{\lambda}_x\}$ of pairwise equivalent finite boundary measures parameterized by points $x\in{\mathcal X}$ (following [@Kaimanovich-Lyubich05] we use the term *stream* for such families) whose mutual Radon–Nikodym derivatives are prescribed by ${\beta}_{\omega}$ in the sense that $$\tag{$**$}
\left| \log\frac{d{\lambda}_x}{d{\lambda}_y}({\omega}) - D{\beta}_{\omega}(x,y)\right| \le C \qquad
\forall\,x,y\in{\mathcal 