Hausdorff measure

see also:

• Hausdorff Dimension

The Hausdorff measure is a method of measuring sets in a metric space that generalizes the concepts of length, area, and volume to non-integer dimensions, playing a crucial role in the study of fractal geometry and the definition of the Hausdorff dimension. It provides a tool for quantifying the “size” of irregular sets, including fractals, which do not fit neatly into the framework of classical geometry.

Formal Definition

The Hausdorff measure is defined for any non-negative real number $d$, where $d$ can be thought of as the dimension. Given a set $S$ in a metric space, the $d$-dimensional Hausdorff measure ${\mathcal{H}}^d(S)$ is constructed as follows:

1. Covering of Set $S$: For any $\delta >0$, consider a countable collection of sets $\{U_i\}$, where each set $U_i$ has a diameter less than or equal to $\delta$, and the union of these sets covers $S$. The diameter of a set, $\text{diam}(U_i)$, is the greatest distance between any two points in $U_i$.

2. Sum of Diameters Raised to $d$: Calculate the sum of the $d$th powers of the diameters of these covering sets: ${\Sigma }_i(\text{diam}(U_i){)}^d$

3. Infimum Over All Coverings: For a fixed $\delta$, find the infimum (the greatest lower bound) of the sums calculated in step 2 over all possible $\delta$-coverings of $S$: ${\mathcal{H}}_{\delta }^d(S)=\inf \{{\Sigma }_i(\text{diam}(U_i){)}^d:\text{each }\{U_i\}\text{ is a }\delta \text{-covering of }S\}$

4. Limit as $\delta$ Approaches 0: Finally, take the limit of ${\mathcal{H}}_{\delta }^d(S)$ as $\delta$ approaches 0 to get the $d$-dimensional Hausdorff measure of $S$: ${\mathcal{H}}^d(S)={\lim }_{\delta \to 0}{\mathcal{H}}_{\delta }^d(S)$

Interpretation and Applications

• The Hausdorff measure is a versatile tool that can measure the size of sets that are too irregular for traditional measures. For $d=1,2,3$, it coincides with our classical notions of length, area, and volume, respectively, for sufficiently “nice” sets.
• For non-integer dimensions, the Hausdorff measure captures the scaling and complexity of fractals. For example, the Hausdorff measure of a fractal set in its fractal dimension is neither 0 nor infinite, providing a meaningful measure of its “size.”
• In the context of fractal geometry, the Hausdorff measure helps formalize the concept of fractal dimension, specifically through the Hausdorff dimension, which is defined as the critical value where the Hausdorff measure transitions from infinity to zero.

The Hausdorff measure is a fundamental concept in geometric measure theory, fractal geometry, and analysis on metric spaces. It has been applied in various fields, such as theoretical physics, to describe phenomena that exhibit fractal behavior, and in mathematics, to study the properties of complex sets that arise in various branches of analysis. Its development has enriched our understanding of dimension and measure, extending these ideas beyond their classical confines to accommodate the complexity of fractal sets.