If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, is it true that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$.

Question

If $0\leq f_n$ and $f_n\rightarrow f$ a.e and $\lim\int_Xf_n=\int_X f$, p,rove or disprove that $\lim\int_Ef_n=\int_E f$ for all $E\in\mathcal{M}$.

I think it is true. It is easy to see $\lim\int_Ef_n\geq\int_E f$ using Fatou's Lemma, yet I could not show that $\lim\int_Ef_n\leq\int_E f$.

To disprove, each time I am either frustrated by either a.e convergence or non-negativity.

Thanks in advance,

Answer 1

Take $X= [0,\infty]$ then define take $f_n=n^2 \chi_{[0, \frac{1}{n}]}$ then you $\lim _{n\rightarrow \infty} \int _{X} f_n =\infty$ but $f = 0$ almost everywhere so $\int _{X} f =0$ to fix that define $\tilde{f}_n =n^2 \chi_{[0, \frac{1}{n}]} + \chi _{[1,\infty]} $. Now $\tilde {f} = \chi _{[1, \infty]}$ almost everywhere.

The hypothesis now is satisfied but if you restrict the integral on $[0,1]$ you do not have the desired limiting behavior.