# A covering theorem for singular measures in the Euclidean space

## TL;DR

This paper establishes a covering theorem for singular measures in Euclidean space using Lipschitz slabs, and applies it to analyze the structure of metric currents and their approximation by smooth maps.

## Contribution

It introduces a novel covering theorem for singular measures with Lipschitz slabs and applies it to the study of metric currents and measure approximation.

## Key findings

- Singular measures can be almost entirely covered by families of Lipschitz slabs with arbitrarily small total width.
- Non-absolutely continuous measures can be approximated by smooth maps with controlled Jacobian integrals.
- Every top-dimensional Ambrosio-Kirchheim metric current in uclidean space is a Federer-Fleming flat chain.

## Abstract

We prove that for any singular measure $\mu$ on $\mathbb{R}^n$ it is possible to cover $\mu$-almost every point with $n$ families of Lipschitz slabs of arbitrarily small total width. More precisely, up to a rotation, for every $\delta>0$ there are $n$ countable families of $1$-Lipschitz functions $\{f_i^1\}_{i\in\mathbb{N}},\ldots, \{f_i^n\}_{i\in\mathbb{N}},$ $f_i^j:\{x_j=0\}\subset\mathbb{R}^n\to\mathbb{R}$, and $n$ sequences of positive real numbers $\{\varepsilon_i^1\}_{i\in\mathbb{N}},\ldots, \{\varepsilon_i^n\}_{i\in\mathbb{N}}$ such that, denoting $\hat x_j$ the orthogonal projection of the point $x$ onto $\{x_j=0\}$ and $$I_i^j:=\{x=(x_1,\ldots,x_n)\in \mathbb{R}^n:f_i^j(\hat x_j)-\varepsilon_i^j< x_j< f_i^j(\hat x_j)+\varepsilon_i^j\},$$ it holds $\sum_{i,j}\varepsilon_i^j\leq \delta$ and $\mu(\mathbb{R}^n\setminus\bigcup_{i,j}I_i^j)=0.$ We apply this result to show that, if $\mu$ is not absolutely continuous, it is possible to approximate the identity with a sequence $g_h$ of smooth equi-Lipschitz maps satisfying $$\limsup_{h\to\infty}\int_{\mathbb{R}^n}{\rm{det}}(\nabla g_h) d\mu<\mu(\mathbb{R}^n).$$ From this, we deduce a simple proof of the fact that every top-dimensional Ambrosio-Kirchheim metric current in $\mathbb{R}^n$ is a Federer-Fleming flat chain.