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

For a metric space $X$, let $mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{mathsf FX}$ such that for every $ninmathbb N$ the map $X^ntomathsf FX$, $(x_1,dots,x_n)mapsto {x_1,dots,x_n}$, is non-expanding with respect to the $ell^1$-metric on $X^n$. We study the completion of the metric space $mathsf F^1!X=(mathsf FX,d^1_{mathsf FX})$ and prove that it coincides with the space $mathsf Z^1!X$ of nonempty compact subsets of $X$ 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 $nge 2$ the Euclidean space $mathbb R^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.