Choquet Integrals

Abstract

One of the new development in the Theory of Morrey spaces - as far as this author is concerned - is the use of Hausdorff capacity and its corresponding Choquet Integral $(\int f\;d\,\Lambda ^{d})$ in the development of the predual to $L^{p,\lambda }(\mathbb{R}^{n})$. This we do in the next chapter once we have thoroughly discussed all the tools that are needed to understand and apply these “non-linear” integrals. And it is because $\Lambda ^{d}$ is not a measure (i.e., not countable additive on disjoint sets) that we must carefully define and expose all of the relevant properties of such an integral.