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 d0 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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