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Published online by Cambridge University Press: 22 January 2016
In a Green space Ω we can introduce Martin’s topology and make it the Martin space Ω, Ω is a dense open subset of 
and the kernel
can be extended continuously to 
, where G(p, x) is a Green function in Ω and y0 the fixed point of Ω. is a metric space. 
is divided into two disjoint subsets Δ0, Δ1 and s ∊ Δ1 is characterized by the fact that K(s, x) is a minimal positive harmonic function in x∊Ω.
1) Brelot, M., Choquet, G., Espaces et lignes de Green. Annales Inst. Fourier 3 (1951), pp. 199–263.CrossRefGoogle Scholar
2) Brelot, M., Le problème de Dirichlet. Axiomatique et frontière de Martin. Journal de Math. 35 (1956), pp. 297–335 (pp. 329–330)Google Scholar, Cf. also Martin, R. S., Minimal positive harmonie functions, Trans. Amer. Math. Soc., 49 (1941), pp. 137–172 CrossRefGoogle Scholar. Parreau, M., Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Annales Inst. Fourier 3 (1952), pp. 103–197 CrossRefGoogle Scholar. Naïm, L., Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Annales Inst. Fourier 7 (1957), pp. 183–281.CrossRefGoogle Scholar
3) R. S. Martin, loc. cit., p. 137.
4) R. S. Martin, loc. cit., p. 157.
5) L. Nairn, loc. cit., p. 203 (théorème 3) and p. 205 (théorème 5).
6)
We denote 
, where the metric dist (x, z
0) is the Martin’s metric.
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