Prove or Disprove $f_n\to f$ and $g_n\to g$ and $\int g_n \to \int g$ then $\int f_n$ converges to $\int f$?

Question

Let $(f_n)$ and $(g_n)$ be a sequence of integrable functions with $|f_n|\leq |g_n|$ for all $n$. Moreover, $g_n\to g$ and $f_n\to f$ as $n$ goes to infinity. Also assume $g$ is integrable and $\int_n g_n \to \int g$. Prove or disprove $\lim_{n\to\infty}\int f_n(x) dx= \int f(x) dx$.

I have tried to use dominated convergence but the problem is gn are not bounded necessarily. I've tried to come up with a counterexample where the limit oscillates... but again to no avail. Please help.

Answer 1

EDIT: This was written when the body of the question did not include the assumption $\int g_n \to \int g$.

Let

$$f_n(x)=g_n(x)=\left\{ \begin{array}{lr} n,&x\in[0,1/n)\\ 0,&\text{otherwise} \end{array}\right.$$

Then $f_n=g_n\to0$, and $\int f_n =1$ for all $n$ but $\int f = 0$.

Answer 2

Assuming $g$ is integrable. Just run through the proof of dominated convergence Theorem. By Fatou,

$\int g - \int f = \int (g-f) = \int \liminf_n (g_n-f_n) \le \liminf_n \int (g_n-f_n) = \liminf_n (\int g_n - \int f_n) $ $= \int g - \limsup_n \int f_n$ (since $\int g_n \to \int g)$. So, $\int f \ge \limsup_n \int f_n$. Similarly,

$\int g + \int f = \int (g+f) = \int \liminf_n (g_n+f_n) \le \liminf_n \int (g_n+f_n) = \liminf_n (\int g_n + \int f_n)$ $= \int g + \liminf_n \int f_n$. Therefore, $\liminf_n \int f_n \ge \int f$.