Nodal domain theorems à la Courant.
We study questions related to exceptional sets of pluri-Green potentials in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials are defined by , where for a fixed z ∈ B, denotes the holomorphic automorphism of B satisfying , and for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of B, then ...
We study the relation between the Lusin-Menchoff property and the -“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the -“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.