I have seen in several papers the claim that for a compact Riemannian manifold the Hausdorff measure will be independent of the Riemannian metric chosen.
Could someone explain what this means exactly and why it is true? I would be inclined to believe that this is the case for the Hausdorff dimension but I don't understand why it would be true for the measure?
Maybe I am missing something obvious!
The statement is obviously false as it stands. If you multiply distances by a constant factor $c$, you multiply $d$-dimensional Hausdorff measure by a constant factor $c^d$.