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We prove that for any singular measure $\mu$ on $\mathbb{R}^n$ it is possibleto cover $\mu$-almost every point with $n$ families of Lipschitz slabs ofarbitrarily 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}},$$f_i^j:\{x_j=0\}\subset\mathbb{R}^n\to\mathbb{R}$, and $n$ sequences ofpositive real numbers $\{\varepsilon_i^1\}_{i\in\mathbb{N}},\ldots,\{\varepsilon_i^n\}_{i\in\mathbb{N}}$ such that, denoting $\hat x_j$ theorthogonal projection of the point $x$ onto $\{x_j=0\}$ and$$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\},$$ it holds$\sum_{i,j}\varepsilon_i^j\leq \delta$ and$\mu(\mathbb{R}^n\setminus\bigcup_{i,j}I_i^j)=0.$  We apply this result to show that, if $\mu$ is not absolutely continuous, itis possible to approximate the identity with a sequence $g_h$ of smoothequi-Lipschitz maps satisfying$$\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 thatevery top-dimensional Ambrosio-Kirchheim metric current in $\mathbb{R}^n$ is aFederer-Fleming flat chain.

A covering theorem for singular measures in the Euclidean space