The completion of the hyperspace of finite subsets, endowed with the ${\mathrm{\ell }}^1$-metric

For a metric space $X$, let $\mathsf{F}X$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d_{\mathsf{F}X}^1$ such that for every $n\in \mathbb{N}$ the map $X^n\to \mathsf{F}X$, $(x_1,\dots,x_n)\mapsto {x_1,\dots,x_n}$,isnon-expandingwithrespecttothe$\ell^1$-metricon$X^n$. We study the completion of the metric space $\mathsf F^1!X=(\mathsf FX,d^1_{\mathsf FX})$andprovethatitcoincideswiththespace$\mathsf Z^1!X$ofnonemptycompactsubsetsof$X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space ${\mathbb{R}}^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.