sufficient condition for mean convergence in $L^1$:prove or disprove Let $f_n \; \colon (0,1) \to \mathbb{R} $ a sequence of Lebesgue integrable functions which converges almost everywhere in $(0,1)$ to $0$. Prove or Disprove: if there exists $p \in (1,+ \infty)$ such that $(f_n)$ is bounded in $L^p(0,1)$,then $\lim_{ n \to \infty} \int_0^1|f_n(x)| \, dx =0$.

For future readers: this answer is a failed attempt at a counterargument.  I’m leaving it here instead of deleting it since I think my faults here can be instructive.  Read the comments below for more detail.
Here’s a modification to what I posted earlier:  let $f_n:(0,1)\to\mathbb{R}$ be defined by
$$
f_n(x) =\begin{cases}
n\ln(n), & \text{if} \;\; x\in\left(0,\frac{1}{n}\right)\\
0, &\text{otherwise}
\end{cases}.
$$
We see that the $f_n\to0$ point wise, however we find that
$$
\int_0^1 |f_n|dx = \frac{1}{n}\cdot n\ln(n) = \ln(n).
$$
