Latest homework question in measure and integration theory course. Suppose that $f_n : \mathbb{R} \to \mathbb{R}$ are integrable, $f_n ≥0$ and $f_n(x) \to f(x)$ for every $x $. Suppose that $\int f_n\to c≥0 .$ Show that $\int f$ exists and $0≤ \int f ≤ c$. Show by examples that every value in $[0 ,c]$ is possible.
I can show $\int f$ exist by showing $\int|f|$ being finite and use Fatou's lemma to show $\int f ≤ \liminf \int f_n = c$. But I can't do the part that $\int f ≥ 0 $. I am also confused since $\int f =\int\lim f_n $, how could I find an example that the Lebesgue integral is not fixed.