25.1.5 Fine topology and balayage3BA364E0394C

Fuglede attributes the classical fine topology to Cartan and the quasi-Lindelöf principle to Doob in [Introduction, p. 227, Fug71Fine]. His preliminaries [Section 1, pp. 228–230, Fug71Fine] are formulated in an axiomatic harmonic space satisfying the domination axiom and admitting a positive potential, and they assume the general duality theory of sweeping before fine connectedness is studied. In the Euclidean setting of this book the quasi-Lindelöf principle, theorem 4.2.22, follows directly from Choquet's lemma and the polarity of the exceptional set of an upper envelope. By example 4.2.8, the fine topology is not Lindelöf, so a countable subfamily can only be expected to cover the union of a family of finely open sets up to a polar set.

Reduction and balayage (theorem 4.2.14, theorem 4.2.16, theorem 4.2.18) are treated only on a ball, where the Green function is explicit and symmetric and Green energy is a Dirichlet integral by proposition 4.2.11. In this setting the swept measure is determined by an orthogonality relation for the energy, which also gives additivity, and the general duality of sweeping recalled by Fuglede is not required. Example 4.2.20 expresses the swept potential of a Dirac measure through harmonic measure, for a compact set whose complement in the ball is a regular domain. Proposition 4.2.25 is proved directly from the definitions; it shows that for $n\geq 2$ the classical fine topology of $\mathbb C^ n=\mathbb R^{2n}$ differs from the topology generated by the upper semicontinuous functions that are subharmonic or identically $-\infty $ on complex lines, which is the plurifine topology of section 5.47 (example 5.47.17).