On the theory of moduli of surfaces - Ryazanov, V.; Sevost’ yanov, Ye. (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 517.5We continue the development of the theory of moduli of the families of surfaces, in particular, strings of various dimensions $m=1,2,…,n-1$ in Euclidean spaces $\mathbb{R}^n,$ $n≥ 2.$ On the basis of the proof of Lemma 1 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 known Väisälä theorem for families of curves to the families of surfaces of arbitrary dimensions. It should be emphasized that the crucial place in the proof of Lemma 1 is Proposition 1 on measurable (Borel) hulls of sets in Euclidean spaces. In addition, we prove similar Lemma 2 and Proposition 2 for the families of concenteric spheres.