I'm dealing with this problem.
Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and $\displaystyle\int_\Omega f_nd\mu\xrightarrow{n\rightarrow\infty}\int_\Omega f_0d\mu<\infty$. Prove that $\displaystyle\int_{\Omega}|f_n-f_0|d\mu\xrightarrow{n\rightarrow\infty} 0$.
If the convergence of $\{f_n\}$s to $f_0$ were almost everywhere, then the problem would become Scheffé's lemma.
By convergence in measure, for each $\epsilon>0$ and natural $n$, define : $\displaystyle E^\epsilon_n:=\{\omega\in\Omega\;;\;|f_n-f_0|(\omega)\geq\epsilon\} \;\wedge\;F_n^\epsilon:=\Omega-E^\epsilon_n$ $$\int_\Omega |f_n-f_0|d\mu=I_1+I_2\quad;\quad I_1=\int_{E^\epsilon_n} |f_n-f_0|d\mu\;\wedge\;I_2=\int_{F^\epsilon_n} |f_n-f_0|d\mu$$ We have $\mu(E_n^\epsilon)\rightarrow 0$ and $|f_n-f_0|<\epsilon$ over $F_n^\epsilon$. But I can't control the terms $|f_n-f_0|$ in $I_1$ and $\mu(F_n^\epsilon)$ in $I_2$ when the measure of space is not finite !
EDIT
My book has never mentioned "Scheffé's lemma".