Summary: We establish that the weighted \(p\)-capacity is equal to the weighted \(p\)-modulus for a condenser \((F_0,F_1,G/E)\), \(1<p<\infty\), under very general assumptions about the topology of \(F_0,F_1,G\), and \(E\) in the compactified Euclidean space \(\overline \mathbb{R}^n= \mathbb{R}^n\cup \{\infty\}\). In particular, this equality yields assertions about the “continuity” of the \(p\)-modulus in the sense of M. Ohtsuka [cf. Analytic Functions, Błażejewko, 1982, Lect. Notes Math. 1039 (1983) p. 467 problems section], and about the \(p\)-capacity and the \(p\)-modulus of a condenser in the sense of M. Brelot [Éléments de la théorie classique du potentiel, 3rd ed., Paris, 1959; Zbl 0084.30903)].
Moreover, we introduce weighted \(p\)-exceptional sets in the sense of B. Fuglede [Acta Math. 98, 171-219 (1957; Zbl 0079.27703)] and give an application of them to the theory of function spaces.