Show that $\int_{E} \vert f_{k} − f \vert dm \to 0.$

Question

Problem: Let $f_k$ be a sequence of non-negative measurable functions such that $f_k \to f$ a.e on $E,\; f \in L(E)$ and $\int_{E} f_{k}dm \to\int_{E} f dm$. Show that $\int_{E} \vert f_{k} − f \vert dm \to 0.$

What I know: I cant use the Scheffe's Lemma because I don't know $f_{k}$ are integrable or bounded. So I can't use the $g_{k}=\vert f_{k}-f\vert\leq f_{k}+f$ and consequently DCT.

I need just a hint, not the whole solution.

Answer 1

By Fatou's lemma, \begin{align*} \int 2f = \int \lim_{k\to\infty} \left(f_k+f - \lvert f_k-f\rvert\right) &\leq \liminf_{k\to\infty} \int\left(f_k+f - \lvert f_k-f\rvert\right)\\ &\leq \int 2f - \limsup_{k\to\infty} \int\lvert f_k-f\rvert \end{align*} Thus, $$ 0\leq\limsup_{k\to\infty} \int\lvert f_k-f\rvert \leq \int 2f - \int 2f = 0 $$ Which proves the result.

Answer 2

For large enough $k$, $f_{k}$ is integrable. Using the reverse Fatou's Lemma (which requires precisely this condition on $f_{k}$), we have

$\lim_{k\to\infty}\int_{E} |f_{k} - f|dm \leq \lim_{k \to \infty}\sup \int_{E}|f_{k} - f|dm \leq \int_{E} \lim_{k \to \infty}\sup |f_{k} - f|dm = 0$.