关于极集的Evans定理和Choquet定理
On Evans' and Choquet's theorems on polar sets
摘要：
根据G.C. Evans和G. Choquet在“势论中的好核$G$“中，对每个极$K_σ$ -集$P$，存在$P$上的有限测度$μ$使得$Gμ=\infty$在$P$上，并且一个集$P$允许$μ$在$P$上的有限测度$μ$使得$\{Gμ=\infty\}=P$当且仅当$P$是极$G_δ$ -集. Evans定理的一个已知应用是用Perron-Wiener-Brelot方法只使用调和上下函数得到开集广义Dirichlet问题的解。在这份说明中，它表明，通过基本的“度量”的考虑，而不使用任何潜在的理论，这样的结果可以得到一般的内核$G$满足当地的三角性质。特殊情况下，在$R^d$上的$G(x,y)=|x-y|^{α-d}$，$2<α<d$，解决了一个长期未决的问题。
英文摘要：
By classical results of G.C. Evans and G. Choquet on "good kernels $G$ in potential theory", for every polar $K_σ$-set $P$, there exists a finite measure $μ$ on $P$ such that $Gμ=\infty$ on $P$, and a set $P$ admits a finite measure $μ$ on $P$ such that $\{Gμ=\infty\}=P$ if and only if $P$ is a polar $G_δ$-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. In this note it is shown that, by elementary "metric" considerations and without using any potential theory, such results can be obtained for general kernels $G$ satisfying a local triangle property. The particular case, $G(x,y)=|x-y|^{α-d}$ on $R^d$, $2<α<d$, solves a long-standing open problem.
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