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

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