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This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to...

If $B_{\alpha ,p}$ denotes the Bessel capacity of subsets of Euclidean $n$-space, $\alpha \>0$, $1\<p\<\infty$, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1\<\alpha p\le n$, there is a constant $C=C(\alpha ,p,n)$ such that for any set $E$ $$min\{B_{\alpha ,p}(E\cap Q),B_{\alpha ,p}(E^c\cap Q\left)\right\}\le C·B_{\alpha ,p}(Q\cap {\partial }_{f}E)$$ for all open cubes $Q$ in $n$-space. Here ${\partial }_{f}E$ is the boundary of the $E$ in the $(\alpha ,p)$-fine topology i.e. the smallest topology on $c$-space that makes the associated $(\alpha ,p)$-linear potentials continuous there. As a consequence, we deduce that...