On the completion of Skorokhod space (Q2201545)

The authors give an explicit description of the completion of the (classical) Skorokhod space [\textit{A. V. Skorokhod}, Teor. Veroyatn. Primen. 1, 289--319 (1956; Zbl 0074.33802)] of càdlàg functions from the unit interval \([0,1]\) to itself. Let \(\mathbb{D}\) denote this Skorokhod space and let \(\mathbb{D}^+\) be the product \(\mathbb{D}\times\Sigma\), where \(\Sigma\) denotes the set of all continuous non-decreasing functions from \([0,1]\) to itself. The distance between two pairs \(\langle F,\sigma\rangle\) and \(\langle G,\tau\rangle\) is defined to be the infimum of all sums \[ \|F\circ\gamma-G\|+\|\sigma\circ\gamma-\tau\| \] where \(\|\cdot\|\) denotes the uniform norm and \(\gamma\) runs through all order-preserving homeomorphisms of~\([0,1]\). This defines a pseudometric. The map \(F\mapsto\langle F,\iota\rangle\) is an isometric embedding of~\(\mathbb{D}\) into~\(\mathbb{D}^+\) and so the latter's metric quotient is the desired completion.