A decomposition for Borel measures $\mu \leq \mathcal{H}^s$
(2022) Real Analysis Exchange — Vol. 48, n° 1, p. 1-17 (2022)
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- 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 $\mu|_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}^s_\infty$. 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 give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.
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Detaille, A., & Ponce, A. (2022). A decomposition for Borel measures $\mu \leq \mathcal{H}^s$. Real Analysis Exchange, 48(1), 1-17. https://doi.org/10.14321/realanalexch.48.1.1629953964 (Original work published 2022)