Measure Theory

Measure theory provides the language of size and integration underlying probability and modern analysis.

Geometric measure theory, the largest field, is mostly the Falconer distance set problem in the plane: the infimum of the \(s\) such that every compact \(E \subseteq \mathbb{R}^2\) of Hausdorff dimension greater than \(s\) has a distance set of positive Lebesgue measure is at least \(1\) and at most \(3/2\) (both Falconer), \(4/3\) (Wolff) and \(5/4\) (Guth, Iosevich, Ou and Wang). By Moran's formula, a self-similar set generated by \(n\) similarities of ratio \(\lambda\) under the open set condition has Hausdorff dimension \(\log n / \log(1/\lambda)\). Measures and measurable sets run from Carathéodory's extension theorem to Vitali's nonmeasurable set and the Banach–Tarski paradox, and integration, Lebesgue spaces, Haar measure and optimal transport make up most of the rest.

• Measures and measurable sets
• Lebesgue measure
• Haar measure and fundamental domains
• Integration and convergence
• Lebesgue spaces
• Differentiation of real functions
• Optimal transport
• Geometric measure theory

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