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.

[ariannaricci]

Thank you, your explanations are very clear fr!

[brendawilliams]

Sometimes it amazes me how a flat sheet of paper can morph numbers right before your eyes. Interesting. Thx.

[asmakamoum]

Hi, please could you give me some titles of book that talk about hausdorff measure.
Thanks a lot.