Geometric Measure Theory: Tangent Measures and Marstrand Theorem

Mikołaj Pabiszczak (UAM)

Geometric Measure Theory is concerned with studying properties of sets with the tools of measure theory. One of the questions that led to its development asks whether infinitesimal properties of measure determine the structure of its support.

One of the results concerning that problem is due to Marstrand. He proved that if \(\mu\) is a Radon measure on \(\mathbb{R}^n\), \(\alpha\) is a nonnegative real number and \(E\) is a Borel set with \(\mu(E)>0\) and if moreover \[ 0< \theta^{\alpha}(\mu,x):=\lim\limits_{r\downarrow 0}\frac{\mu(B_r(x))}{\omega_\alpha r^\alpha} <\infty \quad \text{for \(\mu\)-a.e. \(x\in E\),}\] where \(\omega_\alpha\) is the volume of an \(\alpha\)-dimensional unit ball and \(B_r(x)\) is an open ball of radius \(r\) centred at \(x\), then \(\alpha\) is an integer.

I shall use this theorem as an excuse to introduce the notion of tangent measure and sketch a proof of Marstrand theorem that uses certain properties of tangent measures (proving some of those properties), e.g. the locality of the set of tangent measures or that tangent measures preserve so called \(\alpha\)-uniform measures.