Hi all,

I would like to prove rigorously that the fractal dimension of the unit interval is one. Indeed, it feels obvious from the definition of the Hausdorff Measure that at dimension 1, it measures length.

It is relatively easy to prove that the Hausdorff measure is zero for dimensions greater than 1, but showing the Hausdorff Measure is infinity for dimension less than 1 and indeed 1 when looking at dimension equal to one is proving more difficult.

Thanks in advance!