integration of limit of nonnegative measurable function

Question

$\{f_n\}$are nonnegative measurable function of Lebesgue measure,and a.e.convergence to f,and $$\lim_{n\rightarrow\infty}\int_X f_n dx=\int_X f dx$$ noted by $a$,prove that for $\forall E\subset X$,$$\lim_{n\rightarrow\infty}\int_E f_n dx=\int_E f dx$$ if $a<\infty$. And give counterexample for $a=\infty$.

I have proved the result by Fatou lemma, but i have thouble in finding counterexample.

Any idea will be helpful.

Answer 1

On $X=(0,\infty),$ define

$$f_n(x) = \begin{cases} n^2x^n,& 0<x<1\\ 1,& x\in[1,\infty)\end{cases}$$

Define $f=\chi_{[1,\infty)}.$ Then $f_n\to f$ pointwise everywhere, and $\int_0^\infty f_n = \int_0^\infty f=\infty$ for all $n.$ But take $E=(0,1).$ Then we have $\int_E f_n\to\infty,$ while $\int_E f=0.$

Answer 2

$X=R$,define\begin{equation} f_n(x)= \left\{ \begin{aligned} &n+1 \quad x\in(0,1/n) \\ &1 \quad \mbox{otherwise} \end{aligned} \right. \end{equation} $f_n\rightarrow 1$ pointwise, put $E=(0,1)$.