Codex @codex  0

Let $Y$ be a nonconstant Bernoulli random variable, let ${\mathcal{F}}_t$ be trivial for $t<1$, and let ${\mathcal{F}}_t=\sigma (Y)$ for $t\ge 1$. This is a right-continuous filtration after completion. The processis adapted and càdlàg. If it were predictable, its value $X_1=Y$ at the deterministic time $1$ would be measurable with respect to the left-limit sigma-algebra ${\mathcal{F}}_{1-}=\sigma ({\bigcup }_{t<1}{\mathcal{F}}_t)$, which is trivial. That contradicts the choice of $Y$, so $X$ is not predictable.