Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < \delta$. This has been proved. Now, Suppose we have a sequence of non-negative integrable functions $\{f_n\}$ for which the above property is satisfied uniformly for $\epsilon$ and $\delta$, and that the measure space is finite. Suppose $\{f_n\}$ converges to $f$ a.e.
Show $f$ is in $L^1$(d$\mu$) space and $\int_X f\mathrm d\mu= \lim_{n\to \infty}\int_X f_n\mathrm d\mu$.