Convergence of integral of sequence of functions and its absolute value

Question

Let $f_n$ be a sequence of Lebesgue integrable functions on a measurable subset $\Omega \subset \mathbb{R}$ such that \begin{eqnarray}
\int\limits_{\Omega}f_n \rightarrow \int\limits_{\Omega}f.
\end{eqnarray}
Then, does it imply
\begin{eqnarray}
\int\limits_{\Omega}|f_n| \rightarrow \int\limits_{\Omega}|f|?
\end{eqnarray}
If so, how to prove it? If not, what is the counter example.

Answer 1

Take $\Omega := (-1,1)$, $f(t)=0$ and $f_n (t)=t$ for every $t \in \Omega$ and $n \in \mathbb{N}$. Clearly, $$\int_{\Omega} f_n = \int_{\Omega} f = 0$$ for every $n \in \mathbb{N}$. However, $$\int_{\Omega} |f_n| = \int_{-1}^{1} |t| \mathrm{d}t =1.$$