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.

By default avoid non-ASCII characters.

CB
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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