It is well known that the Hausdorff dimension of the Koch curve is $\log_34\approx1.26>1$. Thus by the property of Hausdorff measure, $\forall1<p<1.2$, the $p$-Hausdorff measure of the Koch curve is $\infty$. On the other hand, the Koch curve consists of countably many line segments, whose $p$-Hausdorff measure is $0$, leading to the conclusion $p$-Hausdorff measure of the Koch curve is $0$. It seems to be a contradiction. Am I missing something? I am confused by my thoughts.
2026-03-25 22:04:35.1774476275
Hausdorff dimension of the Koch curve
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