but showing the Hausdorff Measure is infinity for dimension less than 1...

Consider just the half interval .

Let's suppose that is finite for some . Choose a covering of costisting of intervals of individual length . Their number must be no less than (as is also a covering for the Lebesgue measure of I in ). This means that the related Hausdorff sum will be equal to , where is a constant not depending on or . So, the infimum of all such partitions (which we assumed exists) satisfies . Now in the last relation, the limit of the right hand side as behaves like the limit of as , which is infinite. A contradiction.