[@Tukia97] proved [Theorem \[th:main\]]{} for Kleinian groups with respect to an arbitrary invariant conformal stream.

Our completely elementary approach subsumes all these particular cases and identifies the conservative part of the boundary action with the big horospheric limit set in full generality, for an arbitrary invariant quasi-conformal stream on a general Gromov hyperbolic space.

It is clear from looking at criterion ($*$) that the right object in the context of studying conservativity of boundary actions is the big horospheric limit set ${\Lambda}^{\mathsf{horB}}$ rather than the small one ${\Lambda}^{\mathsf{horS}}$. Nonetheless it is plausible that ${\Lambda}^{\mathsf{horB}}\setminus{\Lambda}^{\mathsf{horS}}$ is a null set with respect to any invariant quasi-conformal stream on an arbitrary Gromov hyperbolic space. This was proved by Sullivan [@Sullivan81] for Kleinian groups with respect to the visual stream, and for subgroups of a free group (again with respect to the uniform stream) it was done in [@Grigorchuk-Kaimanovich-Nagnibeda07].

We refer the reader to [@Grigorchuk-Kaimanovich-Nagnibeda07] for a recent detailed study of the interrelations between various kinds of limit sets in the simplest model case of the action of a free subgroup on the boundary of the ambient finitely generated free group. Actually, it was this collaboration that brought me to the issues discussed in 