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 its potential Gμ is infinite on P, and a set P admits a finite measure μ on P such that Gμ is infinite exactly on 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. It is shown that, by an elementary “metric sweeping” of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = |x − y|^α−d on \({\mathbbm {R}^{d}}\), 2 < α < d, solves a long-standing open problem.

通过对“好” GC Evans和G. Choquet模糊的经典结果内核摹潜在的理论，对于每个极ķ _σ -set P，存在一个有限测度μ对P，使得其潜在摹μ是无限的P，和集P承认一个有限测度μ上P使得ģ μ是无限的恰好在P当且仅当P是一个极性ģ _δ-放。埃文斯定理的一个已知应用通过仅使用谐波上下函数的Perron-Wiener-Brelot方法得出了针对开放集的广义Dirichlet问题的解。结果表明，通过对度量进行基本的“度量扫描”而不使用任何潜在的理论，对于满足局部三角形特性的一般内核G可以得到这样的结果，该属性G局部等于G的某些负幂一些指标。特殊情况下，G（x，y）= | x − y | ^{α - d}上\（{\ mathbbm {R} ^ {d}} \），2 <α < d，解决了一个长期存在的开放问题。