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Conditions for equality of Hausdorff and packing measures on \(\mathbb R^ n\). (English) Zbl 0879.28014

For any \(h:(0, \infty) \rightarrow (0, \infty)\) nondecreasing with \(h(0+)=0\) the \(h\)-Hausdorff measure is compared with the \(h\)-packing measure on \(\mathbb R^n\). They are found to be positive and finite and to coincide on some subset \(A\) of \(\mathbb R^n\) if and only if \(\lim_{r\searrow 0} h(tr) / h(r)\) exists for each \(t > 0\) and \(h\) is a density function, i.e., for some measure \(\mu\) on \(\mathbb R^n\) the limit \(\lim_{r \searrow 0} \mu B(x, r) / h(2r)\) exists and belongs to \((0, \infty)\) for \(\mu\) almost every \(x\in \mathbb R^n\).

MSC:

28A78 Hausdorff and packing measures