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Simplified Construction of Integer Dimension Hausdorff Measures

Luis A. Cedeño-Pérez

Abstract

A simplified construction of the integer dimension Hausdorff measures is given applying methods of geometry and generalized convergence. This construction prioritizes integration, yields the Area Formula as a byproduct of the construction and the Coarea Formula follows indirectly from the smooth case. Though this construction is not as general as the traditional one, it provides a much simpler introduction to the topic and is enough for certain applications to Geometric Analysis.

Simplified Construction of Integer Dimension Hausdorff Measures

Abstract

A simplified construction of the integer dimension Hausdorff measures is given applying methods of geometry and generalized convergence. This construction prioritizes integration, yields the Area Formula as a byproduct of the construction and the Coarea Formula follows indirectly from the smooth case. Though this construction is not as general as the traditional one, it provides a much simpler introduction to the topic and is enough for certain applications to Geometric Analysis.

Paper Structure

This paper contains 13 sections, 19 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. A motivating problem
  3. Hausdorff Measure
  4. Summary of the Construction
  5. Generalized Convergence
  6. Generalized Limits of Measures
  7. Geometry of Volumes
  8. Integer Dimension Hausdorff Measures
  9. Construction and Area Formula
  10. Coarea Formula
  11. Comparison with Hausdorff measure
  12. Riemannian Setting
  13. Concluding Remarks

Key Result

Lemma 2.0.1

Let $(x_{i})_{i\in I}$ be an increasing net of real numbers. If $(x_{i})_{i\in I}$ is upper bounded then

Theorems & Definitions (40)

  • Definition
  • Definition
  • Definition
  • Lemma 2.0.1
  • proof
  • Definition
  • Lemma 2.0.2
  • proof
  • Proposition 2.0.1
  • Proposition 2.0.2
  • ...and 30 more