On random measures, unordered sums and discontinuities 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 \textit{rigorously} the representation and meaning of sums of jumps of type $\sum_{0 < s \leq t} \Phi \circ | \Delta X_s |$, where $X : \Omega \times \R_+ \longrightarrow \R$ is a stochastic process with regulated trajectories, $t \in \R_+$ and $\Phi : \R_+ \longrightarrow \R_+$ is a strictly increasing function which maps 0 to 0 (cf. Proposition \ref{prop:sum of jumps on R+ with invertible function}). Moreover, our approach enables a natural extension of the jump measure of c\`{a}dl\`{a}g and adapted processes to an integer-valued random measure of optional processes with regulated trajectories which need not necessarily to be right- or left-continuous (cf. Theorem \ref{thm:optional random measures}). 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 \ref{lemma:right limits and left limits}, Theorem \ref{thm:at most countably many jumps on compact intervals} and Theorem \ref{thm:at most countably many jumps on R+}). 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 \ref{thm:at most countably many jumps on compact intervals} nor a precise definition of sums of jumps seems to be available yet.