Let $(\Omega, \mathscr F, \mu)$ be a measure space, and $\mathscr G$ be a sub $\sigma$-field of $\mathscr F$.
And let $f:\Omega \to \mathbf R$ be a function, and $g:\Omega \to \mathbf R$ be a $\mathscr G$-measurable function such that
$$f=g \;\text{ $\mu$- a.e.}$$
Then, is $f $ a $\overline{\mathscr G} $-measurable function ?, where
$$ \overline{\mathscr G} = \{A \subset \Omega \mid \exists B \in \mathscr G, A \triangle B \text{ is a $\mu$-null set }\} .$$
I know it is true that $f $ is $\hat{\mathscr G }$-measurable, where $\hat{\mathscr G }= \{A \subset \Omega \mid \exists B \in \mathscr F, A \triangle B \text{ is a $\mu$-null set }\}$.
The answer is yes. If $I$ is an interval let $C$ and $D$ be the inverse images under $f$ and $g$. Then $C\Delta D$ is a $\mu$ null set and $D$ is in $\mathscr G$. The desired measurability follows.