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.
Similar content being viewed by others
References
B. T. Rushing, Topological Embeddings, Academic Press, New York (1973).
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992).
H. Federer, Geometric Measure Theory, Springer, Berlin (1969).
S. Saks, Theory of the Integral, Dover, New York (1964).
T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin (1955).
B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
D. Kovtonyk and V. Ryazanov, “On the theory of mappings with finite area distortion,” J. Anal. Math., 104, 291–306 (2008).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Sci.+Business Media, LLC, New York (2009).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, “Mappings with finite length distortion,” J. Anal. Math., 93, 215–236 (2004).
B. Bojarski, V. Gutlyanskii, O. Martio, and V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, European Mathematical Society, Zürich (2013).
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation. A Geometric Approach, Springer, New York (2012).
E. A. Sevost’yanov, “On the boundary and global behavior of mappings of Riemannian surfaces,” Filomat, 36, 1295–1327 (2022).
P. R. Halmos, Measure Theory, Springer, New York (1974).
B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Corrected Reprint of the 2nd (1965) edition, Dover Publ., Inc. Mineola, NY (2003).
K. Kuratowski, Topology, Vol. 1, Academic Press, New York (1968).
D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, On the Theory of Mappings of the Sobolev and Orlicz–Sobolev Classes [in Russian], Naukova Dumka, Kiev (2013).
E. A. Sevost’yanov, “On the local and boundary behavior of mappings of factor spaces,” Complex Var. Elliptic Equat., 67, No. 2, 284–314 (2022).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 9, pp. 1267–1275, September, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i9.7651.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ryazanov, V., Sevost’yanov, E. On the Theory of Moduli Of The Surfaces. Ukr Math J 75, 1443–1452 (2024). https://doi.org/10.1007/s11253-024-02271-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-024-02271-5