We show that any càdlàg predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated "from below" by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.
"On a dyadic approximation of predictable processes of finite variation." Electron. Commun. Probab. 19 1 - 12, 2014. https://doi.org/10.1214/ECP.v19-2972