文摘
In this paper, we extend the well-known Frostman lemma by showing that for any subset \(E\) of \([0, 1]\) and \(\alpha >0\), if the \(\alpha \)-Hausdorff measure of \(E\) is positive, then there exist a non-zero Borel measure \(\mu \) on \([0, 1]\), a constant \(C>0\) and a subset \(E_0\) of \(E\) such that \(\mu (I) \le C \vert I \vert ^{\alpha }\) for any interval \(I\) and \(E_0\) is dense in the support of \(\mu \). Under an additional condition on \(E_0\), we show that \(\mu (B) = \mu [0, 1]\) for any Borel subset \(B\) containing \(E\). Using the notion of Choquet integral, we extend the notion of capacitarian dimension to arbitrary subset of \([0, 1]\) and prove a generalisation of Frostman’s theorem. Keywords Hausdorff dimension Hausdorff content Frostman lemma Borel measure Capacity Choquet integral