Measure rigidity for leafwise weakly rigid actions

In this paper, given a Borel action $G\curvearrowright X$, we introduce a new approach to obtain classification of conditional measures along a $G$-invariant foliation along which $G$ has a controlled behavior. Given a Borel action $G\curvearrowright X$ over a Lebesgue space $X$ we show that if $G\curvearrowright X$ preserves an invariant system of metrics along a Borel lamination $\mathcal F$, which satisfy a good packing estimative hypothesis, then the ergodic measures preserved by the action are rigid in the sense that the system of conditional measures with respect to the partition $\mathcal F$ are the Hausdorff measures given by the metric system or are supported in a countable number of boundaries of balls. The argument we employ does not require any structure on $G$ other then second-countability and no hyperbolicity on the action as well. Our main result is interesting on its own, but to exemplify its strength and usefulness we show some applications in the context of cocycles over hyperbolic maps and to certain partially hyperbolic maps