Hi I found the following exercise where I'm stuck, I'd appreciate any help. Thanks in advances
Let $0\le f_n\to f$ and $\int f_n\to c>0$ pointwise. Show that $\int \lim f_n d\mu\in [0,c]$ and by an example show that any value in $[0,c]$ is possible.
For the first part: Let $I_n= \int f_n d\mu $. For Fatou's lemma we know that $\int \liminf f_n d\mu\le \liminf I_n$. Since $I_n \to c$, this happens iff $\liminf I_n = \limsup I_n = \lim I_n = c$; similarly, as $f_n\to f$, the limit is the same as the limit inferior and superior for each $x$. For these, it follows that $\int fd\mu$ is defined (as $\int \liminf f_n$ is defined), also $0 \le \int f d\mu \le c$ as desired.
Now for the example I have problems. I have no ideas. The only counterexamples I can think are the trivial where the integral of the limit takes the value of $0$ but the limit of the integral the value of $1$ as in $f_n= 1_{[n,n-1)}$. But I can't figure out one way to get an example where take any value in between $[0,c]$. Any ideas?
You can modify your example a little bit to get the general one (I like to use $\chi_D$ to stand for characteristic function of a set $D$): for any $d\in [0, c]$, let
$$f_n = (c-d) \chi_{[n-1, n]}+ d\chi_{[-2,-1]}.$$