I consider $B_t$ $t\in[0,T]$ a (real valued) stochastic process adapted for the filtration $\mathcal{F}_t$ and $\bar{B}_t$ à modification of $B_t$. I would like to show that $\bar{B}_t$ is adapted for the completion of $\mathcal{F}_t$ that is $\bar{\mathcal{F}}_t$.
Here is my attempt : Let $t\in[0,T]$ and $A$ be a borel set. We know that $\bar{B}_t = B_t$ a.s.
Then we see that $\big\{ \omega\in \Omega : \bar{B}_t(\omega)\in A \big\}\cap\big\{\omega\in \Omega : \bar{B}_t(\omega) = B_t(\omega)\big\}\in\bar{\mathcal{F}}_t$ as it is the set where $\bar{B}_t$ and $B_t$ coincides.
In the other hand $\big\{ \omega\in \Omega : \bar{B}_t(\omega)\in A \big\}\cap\big\{\omega\in \Omega : \bar{B}_t(\omega) \neq B_t(\omega)\big\}$ corresponds to a null set so it is in $\bar{\mathcal{F}}_t$.
So the union of these two sets which is $\big\{ \omega\in \Omega : \bar{B}_t(\omega)\in A \big\}$ is also in $\bar{\mathcal{F}}_t$ since it is a sigma algebra.
I would like to know if it is correct please and if not what can I improve ?
Thank you a lot !