$\{f_n\}$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.
Best Answer
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 $\int_E f=0.$