I am studying how the usual conditions of the filtration are used. Answers in this post In the definition of a Stopping Time, how important are the conditions on the Filtration being complete and right-continuous? nicely explains how they are used for some hitting time or the limit of stopping times to be a stopping time.
Now, when I was reading the answer here https://mathoverflow.net/q/229040, there is a comment saying
Since a martingale is required to be adapted, we'll need completeness of the filtration for the conclusion from the $L^1$ convergence
Could someone elaborate on this?
I think that comment might be mistaken. In general, we would need that the filtration is complete to ensure that $X_n(t) \rightarrow X(t)$ in $L^1$ implies $X(t)$ is $\mathcal F_t$ measurable for each $t$. However, since the answer shows that $X(s) = \mathbb{E}[X(t)|\mathcal F_s]$, we can use the fact that $\mathbb{E}[X(t)|\mathcal F_s]$ is $\mathcal F_s$ measurable to conclude $X(s)$ is $\mathcal F_s$ measurable.