$\int_\Omega\liminf_{h\rightarrow\infty}k\vert f_h\vert^p$, where $\{f_h\}_{h\in\mathbb{N}}$ converges in $ L^p$

Question

I have a problem in which I have to prove l.s.c. of a certain functional. After a couple of inequalities I still have to show $$\int_\Omega\liminf_{h\rightarrow\infty}\ \ k\vert f_h\vert^p dx=\int_\Omega k\vert f\vert^p dx,$$ where $\Omega\subset\mathbb{R}^n$ is open, k is a constant in $\mathbb{R}$, $\ f\in L^p(\Omega)$ and $\ \{f_h\}_{h\in\mathbb{N}}$ is a sequence in $L^p(\Omega)$ which converges to $f$ in $L^p(\Omega)$. I know that convergence in $L^p$ implies convergence almost everywhere of a subsequence, which means $\exists\ f_{h_{j}}$, s.t. $f_{h_{j}}(x)\rightarrow f(x)$ for almost every x in $\Omega$, but I'm not sure how to combine this information with the $\liminf$. I have to admit that I'm not sure about the equality, for my purpouse it's enough the inequality $\geq$. Any hint is appreciated.

Answer 1

Hint: On $[0,1]$ consider the sequence

$$1-\chi_{[0,1/2]},\, 1-\chi_{[1/2,1]}, 1-\chi_{[0,1/3]},\,1-\chi_{[1/3,2/3]},\,1-\chi_{[2/3,1]},\,\dots$$