How should I show that lim_n ∫〖f_n dm〗 = ∫〖f dm〗?

[Jack3]

Suppose {f_n} is a sequence of functions that converges almost everywhere to a function f
and define F_n = sup_k=1,...n |f_n| .
Show that if the integrals of F_n remain bounded as n goes to infinity,
then lim_n ∫〖f_n dm〗 = ∫〖f dm〗.

 

[CaptainBlack]

Suppose {f_n} is a sequence of functions that converges almost everywhere to a function f
and define F_n = sup_k=1,...n |f_n| .
Show that if the integrals of F_n remain bounded as n goes to infinity,
then lim_n ∫〖f_n dm〗 = ∫〖f dm〗.

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CB