$X_{T-}$ is $\mathcal{F}_{T-}$ measurable

Question

By definition, $\mathcal{F}_{T-}=\mathcal{F}_0 \vee \sigma(A\cap \{ t<T\}, A \in \mathcal{F}_t, t \in [0,\infty[)$.

Why is $X_{T-}$ is $\mathcal{F}_{T-}$ measurable?

Answer 1

If $Y$ is predictable, then $Y_T$ is ${\cal F}_{T-}$-measurable. The left-limit process $Y$, as defined above, is left-continuous and adapted, hence predictable.

See this source, Lemma 1, for a proof using a monotone class argument.