Is it possible that a set $A$ has Hausdorff dimension $d=\mathrm{dim}_H (A)\in (0, \infty) $ but $ H^d (A)\notin (0, \infty) $? In other words, positive and finite Hausdorff dimension but its Hausdorff measure is never positive and finite?

# Does non-zero and finite Hausdorff dimension imply non-zero and finite Hausdorff measure

dimension-theory-analysisgeometric-measure-theoryhausdorff-measuremeasure-theory

## Best Answer

An example is the real line $\mathbb R$ with its usual metric. Then $$ 1 = \dim_H(\mathbb R),\quad\text{but}\quad H^1(\mathbb R) = +\infty $$

See HERE for an example of a subset $A$ of $\mathbb R$ with $$ 1 = \dim_H(A),\quad\text{but}\quad H^1(A) = 0 . $$