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When you have a sequence of random variables $\{X_n\}$ measurable with respect to some filtration $\{\mathcal F_n\}$, which converges to some random variable $X$ almost surely.

Then what can we say about the measurability of $X$, it is measurable with respect to what $\sigma$-algebra?

Thank you!

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up vote 1 down vote accepted

What we can say is that $X$ is measurable with respect to the completion of the $\sigma$-algebra generated by $\bigcup_{n\geqslant 1}\mathcal F_n$.

In general, we can't hope better. For example, if for each $n$, there is $A_n\in\mathcal F_n\setminus \mathcal F_{n-1}$, then take $X_n:=\sum_{j=1}^n3^{-j}\chi_{A_j}$. It's a random variable measurable with respect to $\mathcal F_n$ but not $\mathcal F_{n-1}$. The limit is measurable for a $\sigma$-algebra containing $\bigcup_{n\geqslant 1}\mathcal F_n$.

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