A covering theorem for singular measures in the Euclidean space (1705.05141v1)

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.