Convergence in $L^1_{loc}(\Omega)$ implies convergence in $L^1(\Omega)$

Question

When working with a problem in Evans' book, I have the following problem:

Assume $\Omega$ is an open set in $\mathbb{R}^N$. Suppose that a sequence of functions $u_n \in L^p(\Omega)$ convergence in $L^1(\omega)$ to $u \in L^p(\Omega)$ for every $\omega$ compactly contained in $\Omega$.

Is it true that $u_n \to u$ in $L^p(\Omega)$?

The converse is easy, but I can't find a proof, or a counterexample for the question.

Thanks for your helps.

Answer 1

Take $f_n=\frac1{n^{1/p}} \chi_{[n,2n]}$. Then in any compact set $\omega$ you have that $f_n\to 0$ in $L^p$ or $L^1$ but the sequence does not converge to zero in $L^p(\mathbb{R})$.