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Hausdorff Measure - Why We Need It

All lower dimensional sets are "negligible" in the eye of the Lebesgue measure. But we do need to study these smaller subset in more detail. #Hausdorffmeasure addresses this issue. Every subset has its intrinsic Hausdorff dimension $s$ and the Hausdorff $s$-measure of that set is a nonnegative value that is independent of what ambient #metricspace the set is sitting inside (isometrically). The #Hausdorffdimension of the cantor set in R^1 is log2/log3, definitely bigger than the Hausdorff dimension of a countable set that we will see is zero. Moreover, to define Hausdorff measure, we only need a distance function and no further structure is needed. Thus, the Hausdorff measures provide a tool to do analysis and geometry on non-smooth objects, like fractals.