I was going throught the solutions here: If $f$ is Lebesgue measurable on $[0,1]$ then there exists a Borel measurable function $g$ such that $f=g$ a.e.?
And I have a question on the solution of BBBB. I understand it until the very end. I do not see why $g$ would have to be Borel measurable. After all, the pre-image of ${0}$ would be an $F_\delta$ set with a zero measure set which might not be measurable. What am I missing?