The question was:
Given $\mu$ a positive measure in $(X, \Sigma)$ and $f_n, f:X\rightarrow [0,\infty)$ $\mu$-summable then show that if
$\liminf f_n\geq f$ almost everywhere and $\limsup_n \int_X f_nd\mu \leq \int_X fd\mu$ then $f_n\to f$ in $ L^1$.
The hint was to prove that $g_n=\inf_{k\geq n} f_k$ satisfies $g_n\to f$ in $L^1$> I have done that via Fatou's Lemma and monotone convergence theorem, but I could not infer the main result!