До теорії модулів поверхонь

V. Ryazanov; Ye. Sevost’ yanov

doi:10.3842/umzh.v75i9.7651

V. Ryazanov Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region
Ye. Sevost’ yanov Ivan Franko Zhytomyr State University; Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk, Donetsk region

DOI: https://doi.org/10.3842/umzh.v75i9.7651

Keywords: Moduli of surfaces in Euclidean spaces, Hausdorff measures, measurable (Borel) hulls of sets

Abstract

UDC 517.5

We continue the development of the theory of moduli of the families of surfaces, in particular, strings of various dimensions $m=1,2,\ldots,n-1$ in Euclidean spaces $\mathbb{R}^n,$ $n\geq 2.$ 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.

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On the theory of moduli of surfaces

Abstract

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