A Fubini-type theorem for Hausdorff dimension

TL;DR

This paper establishes a modified Fubini theorem for Hausdorff dimension, showing it holds for most parts of a set when small exceptional sets are removed, with implications for geometric measure theory.

Contribution

It proves a Fubini-type theorem for Hausdorff dimension that accounts for small null sets on Lipschitz graphs, extending classical results to more general sets.

Findings

Fubini theorem for Hausdorff dimension holds modulo small Lipschitz-null sets.

Existence of a null subset G where the Fubini property is valid for the remaining set.

Results extend to Ahlfors-David regular sets and have applications in geometric measure theory.

Abstract

It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz graphs. We say that $G\subset \mathbb{R}^k\times \mathbb{R}^n$ is $\Gamma_k$-null if for every Lipschitz function $f:\mathbb{R}^k\to \mathbb{R}^n$ the set $\{t\in\mathbb{R}^k\,:\,(t,f(t))\in G\}$ has measure zero. We show that for every Borel set $E\subset \mathbb{R}^k\times \mathbb{R}^n$ with $\dim (\text{proj}_{\mathbb{R}^k} E)=k$ there is a $\Gamma_k$-null subset $G\subset E$ such that $$\dim (E\setminus G) = k+\text{ess-}\sup(\dim E_t)$$ where $\text{ess-}\sup(\dim E_t)$ is the essential supremum of the Hausdorff dimension of the vertical sections $\{E_t\}_{t\in \mathbb{R}^k}$ of $E$. In addition, we show that, provided that $E$ is not $\Gamma_k$-null, there is a $\Gamma_k$-null subset $G\subset E$ such that for $F=E \setminus G$, the Fubini-property holds, that is, $\dim (F) = k+\text{ess-}\sup(\dim F_t)$. We also obtain more general results by replacing $\mathbb{R}^k$ by an Ahlfors-David regular set. Applications of our results include Fubini-type results for unions of affine subspaces, connection to the Kakeya conjecture and projection theorems.