Suppose $\{f_n\}$ is a sequence of lebesgue measurable functions such that $f_n\rightarrow f$, except on a set of measure $0$, as $n\rightarrow\infty$, and $|f_n(x)|\leq g(x)$, where $g$ is integrable.
Denote $E_N = \{x: |x| \leq N, g(x)\leq N\}$. If I can prove that $m(E_N^c)\rightarrow\infty$ as $n\rightarrow\infty$, does that tell me anything about the lebesgue integral on that set?
Specifically, can I determine that $\int_{E_N^c} |f_n -f| \leq \epsilon$
for some $\epsilon > 0$, and all large $n$?