If $\|u_n\|_{L^1(\mathbb R)}\leq \gamma $ and $u_n\to u$ a.e., then $u\in L^1(\mathbb R)$ and $\|u\|_{L^1(\mathbb R)}\leq \gamma $.

Question

I'm doing exam of previous year and I'm stuck on this question : If $u_n\in L^1(\mathbb R)$, $\|u_n\|_{L^1(\mathbb R)}\leq \gamma $ (where $\gamma >0$) and $u_n\to u$ a.e., then $u\in L^1(\mathbb R)$ and $\|u\|_{L^1(\mathbb R)}\leq \gamma $.

Try

I just have that $$\|u\|_{L^1(\mathbb R)}\leq \|u-u_n\|_{L^1(\mathbb R)}+\|u_n\|_{L^1(\mathbb R)}\leq \|u_n-n\|_{L^1(\mathbb R)}+\lambda .$$

Now I tried to show that $u_n\to u$ in $L^1$, but unfortunately, I can't. Any idea ?

Answer 1

It's just Fatou's lemma : $$\|u\|_{L^1}=\int_{\mathbb R}\liminf_{n\to \infty }|u_n|\leq \liminf_{n\to \infty }\int_{\mathbb R}|u_n|\leq \gamma .$$