  The goal of this paper is to investigate the tools of extreme value theory
originally introduced for discrete time stationary stochastic processes (time
series), namely the tail process and the tail measure, in the framework of
continuous time stochastic processes with paths in the space $\mathcal{D}$ of
c\`adl\`ag functions indexed by $\mathbb{R}$, endowed with Skorohod's $J_1$
topology. We prove that the essential properties of these objects are
preserved, with some minor (though interesting) differences arising. We first
obtain structural results which provide representation for homogeneous
shift-invariant measures on $\mathcal{D}$ and then study regular variation of
random elements in $\mathcal{D}$. We give practical conditions and study
several examples, recovering and extending known results.
