On the Theory of Moduli Of The Surfaces

Abstract

We continue the development of the theory of moduli of the families of surfaces, in particular, of strings of various dimensions m = 1, 2, . . . ,n − 1 in Euclidean spaces \({\mathbb{R}}^{n}\) , n ≥ 2. On the basis of the proof of the lemma on the relationships between the moduli and Lebesgue measures, we prove the corresponding analog of the Fubini theorem in terms of moduli that extends the well-known Väisälä theorem for the families of curves to the families of surfaces of arbitrary dimensions. It should be emphasized that the crucial role in the proof of the mentioned lemma is played by a proposition on measurable (Borel) hulls of sets in Euclidean spaces. In addition, we also prove a similar lemma and a proposition for the families of concentric balls.