I am aware that this question has been asked many times, but I am still not getting it so I am asking it again.
Given a measure space $(X,M,\mu)$, let $f_n$ be a sequence of complex-valued $\mu$-measurable functions that converge pointwise to $f$ on subset $E$ of $X$ with $\mu(E^c) =0 $. Then, I know that if the measure is not complete, $f$ is not necessarily measurable. But, I read that $f$ can be redefined on $E^c$ so as to become measurable. For instance, one can define $f = 0$ on $E^c$. But, why is then this redefined function measurable?
In Folland's Real Analysis book, he refers to Proposition 2.12 in regards to this matter, but I don't understand how that proposition helps.