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

Abstract

We extend the notion of the variation Vf([a; b]) of a function f : [a; b] → ℝ to the variation Vf(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 ∈ IKi where I is countable and all Ki are compact. Let f : G → ℝm where G ⊂ ℝn with n ≤ m, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < in ≤ m, 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 A ⊂ G 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.