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In the theory of Kleinian groups, Sullivan's classical theorem establishesthe correspondence between Patterson-Sullivan measures and Hausdorff measureson the limit sets for convex cocompact Kleinian groups. This connectionprovides a geometric understanding of Patterson-Sullivan measures, emphasizingtheir association with the internal metric on limit sets. Recent advancementsin the theory of infinite covolume discrete subgroups of higher rank Lie groupshave brought Anosov subgroups into focus as a natural higher rank extension ofconvex cocompact Kleinian groups. This raises an intriguing question: underwhat conditions do Patterson-Sullivan measures of Anosov subgroups emerge asHausdorff measures on limit sets with appropriate metrics?  In this paper, we give a definitive answer to this question by showing that,for all Zariski dense Anosov subgroups, a Patterson-Sullivan measure isproportional to the one-dimensional Hausdorff measure for an intrinsicconformal premetric if and only if it is symmetric. Furthermore, in thesymmetric case, the Patterson-Sullivan measure of any ball is comparable to itsradius.  We also discuss several applications, including analyticity of$(p,q)$-Hausdorff dimensions on the Teichm\"uller spaces and spectralproperties of associated locally symmetric manifolds such as temperedness andthe bottom of the $L^2$-spectrum.

Patterson-Sullivan measures of Anosov groups are Hausdorff measures