Let $\{S_n\}$ denote a simple symmetric random walk starting at $0$, and let $\{\mathcal{F}_n\}$ denote the natural filtration from the sequence of i.i.d. steps of this random walk.
Define the last exit time as $$T= \max\{n\geq 0 : S_n = 10\}.$$ Because $S_n$ is a simple symmetric random walk, then $\mathbb{P}(T = k) = 0$ for each $k \in \mathbb{Z}_{\geq 0}$, because the state $10$ is null recurrent. If we take $\mathcal{F}_k$ to be complete (it contains the probably zero sets) then $\{T = k\} \in \mathcal{F}_k$.
Does this mean that $T$ is actually a stopping time?