Q:

Suppose that $\displaystyle {f_n}_{n=1}^{\infty} \subset L^+ $ , $\displaystyle f_n \longrightarrow f $ pointwise and $\displaystyle \int f = lim \int f_n < \infty $. Prove that $\displaystyle \int_E f = lim \int_E f_n $ for all $\displaystyle E \in M $. Prove this need not be true if $\displaystyle \int f = lim \int f_n = \infty $.

Attempt:

I take any set $\displaystyle E \in M $ (where M is the sigma-algebra of Lebesgue Measurable sets). I know that $\displaystyle \int_E f_n = \int f_n \chi_E $. what can i do from here? can i say that $\displaystyle f_n \chi_E \longrightarrow f \chi_E $? help would be appreciated! thanks