A decomposition for Borel measures $\mu \le \mathcal{H}^{s}$

Abstract

We prove that every finite Borel measure $\mu$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $\mu\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.