The precise statement is if $f$ is integrable, $g$ measurable, and $f = g$ almost everywhere, then $g$ is integrable, and the integrals coincide. I use the following definition of integrable: $f$ is integrable if there is a sequence of (simple) integrable functions $\{f_n\}$ such that $f_n \to f$ almost everywhere, and it is Cauchy in the mean.
Then $\lim_n f_n = f = g$ almost everywhere, and it seems that I can use the same sequence for both $f$ and $g$, and then there is hardly anything to do. What am I missing? Thanks in advance.
Since $f=g$ a.e., thus, as you said: $\lim f_n=g$.
Thus g is measurable and integrable, since it belongs to the closure of the simple functions' span. The two integrals are equal since the functions differs on a zero-measure set (call it A):
$\begin{align} \int_E f=\int_{A/E}f+\int_A f=\\ \int_{A/E}f=\int_{A/E}g=\\ \int_{A/E}g+\int_A g=\int_E g \end{align}$