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Say we have some compact metrisable topological space $X$ with a measure $\mu$ defined on the Borel sets of $X$. Then is there some way to determine whether $\mu$ is the Hausdorff measure associated to some metric $d$ compatible with the topology of $X$? And if so, is there some process to recover a metric from the measure? I'd imagine that there would have to be some conditions placed on the space $X$, eg. that it's connected, and it might even be necessary to assume that it's some nice space such as a manifold, with a "gauge" metric $d_{0}$ relative to whose Hausdorff measure $\mu$ is absolutely continuous, but I'd like to ask the question in the greatest generality possible, in the hope that there is an answer out there.

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I guess you want to say "recover a metric", but not "the metric" --- clearly there are many metrics which give the same Hausdorff measure... – Anton Petrunin Jan 31 '10 at 4:27
I just edited the question to account for your comment, but it leads to another naive question; I can see how a many different metrics on a disconnected set(eg a discrete set) would lead to the same Hausdorff measure, but is there a simple example in the case of connected sets? – Gordon Craig Feb 1 '10 at 4:04
Sure. Just change the standard metric on the real line to $\operatorname{arctan}|x-y|$. All Hausdorff measures associated with power functions will stay exactly the same. – fedja Feb 1 '10 at 4:32

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