A Decomposition for Borel Measures \(\mu \le \mathcal{H}^{s}\)

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