Hausdorff measure

In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rⁿ or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. The one-dimensional Hausdorff measure of a simple curve in Rⁿ is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R² is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory.

Let ${\displaystyle (X,\rho )}$ be a metric space. For any subset ${\displaystyle U\subset X}$, let ${\displaystyle \mathrm {diam} \;U}$ denote its diameter, that is

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