I know the question has been asked but their proofs arent whats on my mind, and I want to know where is the flaw in my proof because I cant seem to find it.
Suppose $\{f_n\}$ are a sequence of non negative measurable functions and $f_n\to f$ point wise, and $\int f=\lim \int f_n<\infty$ show that $\int_E f=\lim \int_E f_n$. where $E$ is measurable.
Update:
We know that $\lim \int f_n=\int f$ then $\lim \sup \int f_n =\lim \inf \int f_n$. Also $\int f=\int_E f+ \int_{E^c} f$ then we have the following, \begin{align*} \int_E f =\int f-\int_{E^c}f&\geq \int f-\lim \sup \int_{E^c} f_n \text{ by fatou lemma} \\&= \lim \sup \int f-\lim \sup \int_{E^c}f_n \\&=\limsup (\int f-\int_{E^c}f_n)=\limsup \int_E f_n \end{align*}
Conversely, by faout's lemma again $$\int_E f=\int f \chi_E \leq \liminf \int f_n \chi_E=\liminf \int_E f$$
Thank you for your help!