Hausdorff Measure and Dimension with Examples

This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for calculating the dimension of a set, ensuring all calculation details are transparent for the reader. The paper first establishes the theoretical groundwork by reviewing the definitions of the Hausdorff measure, proving the dimensional invariance under changes to the shape of the covering sets, and confirming the dimensional property of open sets. It then introduces the two main methodologies. The first is the Lower and Upper Bound Estimation, which uses the relationship between the measure $H^s(A)$ and the dimension $\dim_{H}(A)$. This method emphasizes the use of the Mass Distribution Principle for establishing the lower bound, which is essential when the Lebesgue measure of the set is zero. The second, more computationally efficient method is the Similarity Dimension Method, introduced via the definitions of Similitudes, Iterated Function Systems (IFS), and the Moran-Hutchinson Theorem. Both methodologies are applied rigorously to classic examples: the Unit Square (yielding dimension $2$) and the Cantor Set (yielding the fractional dimension $\log 2 / \log 3$). The paper serves as a detailed, step-by-step guide intended to make the application of these fundamental fractal geometry concepts clearer and easier to understand.