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Journal of Mathematical Sciences

, Volume 243, Issue 6, pp 917–921 | Cite as

The Hausdorff Measure on n-Dimensional Manifolds in ℝm and n-Dimensional Variations

  • A. V. PotepunEmail author
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We extend the notion of the variation Vf([a; b]) of a function f : [a; b] →  to the variation Vf(A) of a continuous map f : G → n, where G is an open subset of ℝn, over a set AG of the form A = ∪i ∈ IKi where I is countable and all Ki are compact.

Let f : G → m where G ⊂ ℝn with nm, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < inm, let fα be the map with coordinate functions \( {f}_{i_1},\dots, {f}_{i_n} \). The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝn, and a subset AG has the form A = ∪i ∈ IKi where I is countable and all Ki are compact, then \( {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) \) where \( {V}_{f_{\alpha }}(A) \) is the variation of fα over A and Hn is the n-dimensional Hausdorff measure in ℝm.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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