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Assume we can decompose a set $A$ in $\mathbb{R^n}$ of Hausdorff-dimension n into sets $(A_t)$ $t\in [0,1]$ of Hausdorff-dimension n-1 whose n-1-dimensional volume is known (for example is zero).

Are there any possibilities beside the coarea-formula that allow us to say something about the measure of A?

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You need to know that $A_t$ depends on $t$ is a reasonable way. Otherewise one can take a bijection from $[0,1]$ to a square, so all your $A_t$ are one-point sets... –  Anton Petrunin Sep 13 '11 at 16:47
    
Figured I'd retag with some arXiv tags. If anyone more familiar with this area sees better tags, please feel free to overrule my choices –  David White Sep 13 '11 at 17:25

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