1 4 Se p 20 06 On random measures , unordered sums and discontinu - ities of the first kind

By investigating in detail discontinuities of the first kind of real-valued functions and the analysis of unordered sums, where the summands are given by values of a positive real-valued function, we develop a measure-theoretical framework which in particular allows us to describe rigorously the representation and meaning of sums of jumps of type ∑ 0<s≤t Φ ◦ |∆Xs|, where X : Ω×R+ −→ R is a stochastic process with regulated trajectories, t ∈ R+ and Φ : R+ −→ R+ is a strictly increasing function which maps 0 to 0 (cf. Proposition 3.13). Moreover, our approach enables a natural extension of the jump measure of càdlàg and adapted processes to an integer-valued random measure of optional processes with regulated trajectories which need not necessarily to be rightor left-continuous (cf. Theorem 4.5). In doing so, we provide a detailed and constructive proof of the fact that the set of all discontinuities of the first kind of a given real-valued function on R is at most countable (cf. Lemma 2.3, Theorem 2.5 and Theorem 2.6). By using the powerful analysis of unordered sums, we hope that our contributions fill an existing gap in the literature, since neither a detailed proof of (the frequently used) Theorem 2.5 nor a precise definition of sums of jumps seems to be available yet.