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.
Categories: math.FA math.AP math.CA