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

Abstract

We extend the notion of the variation V_f([a; b]) of a function f : [a; b] → ℝ to the variation V_f(A) of a continuous map f : G → ℝⁿ, where G is an open subset of ℝⁿ, over a set A ⊂ G of the form A = ∪_i ∈ IK_i where I is countable and all K_i are compact.

Let f : G → ℝ^m where G ⊂ ℝⁿ with n ≤ m, and let f₁, . . . , f_m be the coordinate functions of f. For α = {i₁, . . . , i_n} where 1 ≤ i₁ < i₂ < ⋯ < i_n ≤ m, let f_α 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 A ⊂ G has the form A = ∪_i ∈ IK_i where I is countable and all K_i are compact, then \( {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) \) where \( {V}_{f_{\alpha }}(A) \) is the variation of f_α over A and H_n is the n-dimensional Hausdorff measure in ℝ^m.