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.

Added application to solution of semilinear PDE involving exponential