An Introduction to the Hausdorff Measure and Its Applications in Fractal Geometry

TL;DR

The paper develops the Hausdorff measure $\mathcal{H}^s$ and Hausdorff dimension as central tools in fractal geometry, establishing a complete pipeline from outer measures and Carathéodory measurability to a robust measure-theoretic framework. It presents the general covering-based construction, proves translation invariance and scaling properties, and defines the critical exponent that yields the Hausdorff dimension. The Cantor ternary set is analyzed in depth, yielding $\dim_{\mathcal{H}}(K)=\frac{\log 2}{\log 3}$, Lebesgue measure $\lambda(K)=0$, and sharp bounds $\tfrac{1}{2}\le\mathcal{H}^s(K)\le1$ at the critical dimension, illustrating sharp quantitative fractal behavior. The work thus links abstract geometric measure theory to concrete fractal analysis with broad relevance to mathematics, physics, and computational geometry.

Abstract

This paper presents a comprehensive introduction to the Hausdorff measure, a fundamental tool in fractal geometry and geometric measure theory. We begin by defining the Hausdorff outer measure on subsets of metric spaces, followed by a discussion of Caratheodory's criterion, which characterizes measurable sets. From this foundation, we construct the Hausdorff measure and explore its essential properties, including monotonicity and translation invariance. We then introduce the Hausdorff dimension, a powerful generalization of Euclidean dimension, particularly suited to analyzing non-regular or self-similar sets. As an application, we examine the Cantor ternary set, computing its Hausdorff dimension and demonstrating how the Hausdorff measure captures its geometric complexity. This exposition aims to bridge the gap between abstract theory and illustrative application, offering insights relevant to mathematics and various scientific domains such as physics and computer science.

Figures (2)

• Figure 1: Graph of the function $d\rightarrow \mathcal{H}^d(A)$
• Figure 2: First five construction stages of the Cantor set

Theorems & Definitions (17)

• Remark 1
• Example 1
• proof : Proof of Caratheodory’s Theorem \ref{['th:1']} :
• proof : Proof of Lemma \ref{['lem:1']}:
• proof : Proof of Proposition \ref{['pro:1']}:
• proof : Proof of Theorem \ref{['th:2']}:
• Remark 2
• proof : Proof of Proposition \ref{['pro:2']}:
• proof : Proof of Proposition \ref{['pro:3']}:
• proof : Proof of Proposition \ref{['pro:4']}: