The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on *Stochastic Integration and Differential Equations* at page 80.

And my doubt is that I have not understood yet what does the sets $A = A(\epsilon, t)$ and $B = B(\epsilon, t)$ mean in order to conclude that $A \cup B $ equals the set of stopping times of $X$ on $(0,t]$. The definition of these two sets is the following

For me the definition of $ A= A(\epsilon, t)$ (as a set that depends on $\epsilon $ and $t$) should be: let $A(w) = A(\epsilon, t)(w)$ be any finite set of jump times of $X_{\cdot}(w)$ on $(0, t]$. However, this definition and the definition of $B(w) = B(\epsilon, t)(w)$ as the set of jump times $s$ on $(0,t]$ such that $\sum_{s \in B(\epsilon, t)(w)} (\Delta X_{s} (w))^{2} \leq \epsilon^{2}$, and $A(\epsilon, t)(w) \cup B(\epsilon, t)(w)$ equals the stopping times of $X_{\cdot}(w)$ on $(0,t]$ does not match. Moreover, the definition of $A = A( \epsilon, t)$ does not depend on $\epsilon $, and $A$ and $B$ are not necessarily disjoint according the definition.

Any comment, idea, or hint would be welcome