Intersection properties of directional essential cluster sets (Q1978907)

Let \(H\) be the open upper half-plane and \(W\) be a normal second countable topological space. For a function \(f:H\to W\), \(x\in \partial H\) and \(\alpha \in (0,\pi)\) denote by \(C_e(f,x,\alpha)\) the directional essential cluster set at \(x\) in the direction \(\alpha \) (a natural definition using metric density). Theorem. Let \(f:H\to W\) be a measurable function. Then except possibly at most a countable set of points \(x\in \partial H\), for almost every \(\alpha \in (0,\pi)\) \[ C_e(f,x,\alpha)\cap C_e(f,x,\beta)\neq \emptyset \] for almost every \(\beta \in (0,\pi)\). If, moreover, \(f\) is continuous, then ``almost every'' may be replaced by ``almost every and nearly every'', i.e., except for a measure zero set of the first category.