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

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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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Jack said:
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〗.

Please do not copy and paste text with non-standard characters, not everybodies system will render them correctly, learn to use the LaTeX supported here on MHB and on most other maths boards.

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CB
 
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I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...

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