= Geometric measure theory
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Geometric measure theory uses <measures> and <analysis> to study geometric objects that need not be smooth. Its measuring rules can distinguish a <curve> from its surrounding plane or capture the fine structure of a <fractal>. <Hausdorff measure> supplies size adapted to an exponent, while <Hausdorff dimension> identifies the exponent at which that size changes character.
The <Cantor set> is an elementary starting point: it has zero <Lebesgue measure> but positive <Hausdorff dimension>. In <Hausdorff dimension of the Cantor set>, explicit small <covers> give an upper bound and a distributed <probability measure> supplies the matching lower bound. This combination of geometric coverings and analytic control is useful even before more advanced tools for irregular surfaces are introduced.
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