An elementary proof that the first hitting time of an $F_\sigma$ set by a jump process is a stopping time

TL;DR

This paper provides a simple, elementary proof demonstrating that the first hitting time of an $F_\sigma$ set by a jump process of a càdlàg adapted process is a stopping time.

Contribution

It offers a new, straightforward proof that the first hitting time of an $F_\sigma$ set by such jump processes is a stopping time, simplifying previous arguments.

Findings

First hitting time of $F_\sigma$ set is a stopping time

Elementary proof reduces complexity of previous methods

Applicable to càdlàg adapted processes

Abstract

We give a short and elementary proof that the first hitting time of a $F_\sigma$ set by the jump process of a c\`{a}dl\`{a}g adapted process is a stopping time.