Convergence of Lebesgue integrals I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong:
Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with $\displaystyle\forall x \in E\setminus N: \lim_{k \to \infty}f_k(x)=f(x)$, where $\lambda(N)=0$, $E,N \subset \mathbb{R}^n$ and $\displaystyle\lim_{k \rightarrow \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$.
Is it always true that $$\lim_{k \rightarrow \infty}\int_E |f-f_k(x)|d\lambda=0 ?$$
I see the striking similarity to Lebesgue's dominated convergence theorem, if one could use $\displaystyle\lim_{k \to \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$ to find some majorant $g$ for our $f_k$ it would be true, especially it would be true when the $f_k$ converge against $f$ from below.
 A: We have $f_k$ converges to $f$ pointwise almost everywhere,    $f_k$ and $f$ are non-negative, and $\int  f_k \rightarrow \int  f $.
Note that, since the $f_k$ and $f$ are nonnegative:
$$| f_k-f\,|\le  f_k + f \quad \Rightarrow\quad f_k + f -|f_k-f\,|\ge 0.$$
By Fatou's Lemma:
$$
\tag{1}\liminf_{k\to \infty} \int (  f_k + f -|f_k-f\,|\,)\ \ge \int \liminf_{k\to \infty}  \,(f_k + f -|f_k-f\,|)\,.
$$
Since $\int f_k\rightarrow\int f$, we have
 $$ \tag{2}\liminf_{k\to \infty} \int ( f_k + f -|f_k-f\,|\,)=
2\int  f -\limsup_{k\to \infty}\int|f_k-f\,|. $$
Since $f_k\rightarrow  f$ almost everywhere, we have
$$ \tag{3}\int \liminf_{k\to \infty} \, (f_k + f -|f_k-f\,|\,)= \int  2 f. $$
Substituting the expressions on the right hand sides of (2) and (3) into (1) gives:
$$
2\int  f -\limsup_{k\to \infty}\int|f_k-f\thinspace|\ge\int 2 f\ ;
$$whence
$$
\limsup_{k\to \infty}\int|f_k-f|\le0.
$$
A: Let $g_k^+:=(f_k-f)_+$ and $g_k^-:=(f-f_k)_+$, so $|f-f_k|=g_k^+ + g_k^-$ and $f_k-f=g_k^+ - g_k^-$. First note that $g_k^- \le f$ since $f$ is nonnegative, and $g_k^- \to 0$ pointwise almost everywhere. By dominated convergence, $\int g_k^-\to 0$. But
$$\int (g_k^+ + g_k^-) = \int(f_k-f+2g_k^-) = \int f_k - \int f + 2\int g_k^- \to 0$$
