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On a class of Hausdorff measure of cartesian product sets

Hajer Jebali, Riheb Guedri, Najmeddine Attia

Abstract

In this paper, we study, in a separable metric space, a class of Hausdorff measures $\mathcal{H}_μ^{q, ξ}$ defined using a measure $μ$ and a premeasure $ξ$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures $\mathcal{W}_μ^{q, ξ}$ appeared as an important tool when studying the product sets. When $μ$ and $ξ$ are blanketed, we prove that $\mathcal{H}_μ^{q, ξ} = \mathcal{W}_μ^{q, ξ}$. As an application, the case when $ξ$ is defined as the Hausdorff function is considered.

On a class of Hausdorff measure of cartesian product sets

Abstract

In this paper, we study, in a separable metric space, a class of Hausdorff measures defined using a measure and a premeasure . We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures appeared as an important tool when studying the product sets. When and are blanketed, we prove that . As an application, the case when is defined as the Hausdorff function is considered.

Paper Structure

This paper contains 13 sections, 13 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Construction of generalized measures
  3. Generalized Hausdorff measure
  4. Weighted generalized Hausdorff measure
  5. Regularity of the weighted generalized Hausdorff measure
  6. Proofs of main results
  7. Proof of Theorem A
  8. Proof of Theorem B
  9. Proof of Theorem C
  10. Applications
  11. Application 1 : Definition of $\xi$ using Hausdorff function
  12. Application 2
  13. appendix

Key Result

Theorem 1

Let $\mu\in\mathcal{P}({\mathbb X})$, $q \in\mathbb{R}$ and $\xi \in \Phi({\mathbb X})$. Then

Theorems & Definitions (32)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Lemma 1
  • proof
  • Example 1
  • Theorem 3
  • ...and 22 more