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Is there any co-area formula involving non-integer Hausdorff dimension? Moreover is it sensible to write the following: Let $S$ be a subset in $ \mathbb{R}^n$ with Hausdorff dimension s $(0<s<n)$ and $ S_\epsilon = \bigcup_{x \in S}B(x,\epsilon)$.

$ \int_{S_\epsilon} dy = \int_S \int_{B(x,\epsilon)} \; dy \; d\mu(x)$ where $ \mu $ is a uniformly $s$-dimensional measure.

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