The Skorokhod space

Càdlàg paths on [0,T] with values in a metric space, time changes of [0,T] and Billingsley's J1 metric, with the characterization of J1 convergence by time changes; continuous paths as a closed subspace on which J1 is uniform; evaluation, restriction, constant extension and stopping, continuous at paths continuous at the cut time; time–space scaling, continuous at (f, 1, 1) when f is continuous at the horizon; the largest positive and absolute jumps as 2-Lipschitz functionals; and Billingsley's equivalent complete metric d°: the Skorokhod space of a complete metric space is completely metrizable. For separable E the Skorokhod space is separable, its Borel σ-algebra is generated by evaluations at a dense set of times containing the terminal time, laws are determined by finite-dimensional distributions, and a law has countably many fixed jump times. Convergence in law follows from convergence of the restrictions to [0, a] for a ↑ T plus control of the terminal oscillation, and laws a.s. continuous at T are determined by their restrictions. The càdlàg Vervaat map rotates a path at its first infimum time; it is measurable, matches the discrete Vervaat rotation of words on step paths, and is J1-continuous at paths with a unique interior infimum time. Billingsley's modulus w' gives a compactness criterion and tightness criteria for laws on the Skorokhod space. These results do not depend on trees.