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Given two sequences of integrable functions $\{f_{n}\}, \{g_{n}\}$ with limits $f$ and $g$ both also integrable. Does this always hold

$\lim_{n}(f_{n}-g_{n})=\lim_{n}f_{n}-\lim_{n}g_{n}=f-g$

I mean what if for some point x, $f(x)=\infty$ and $g(x)=\infty$ that would make $f-g=\infty-\infty$. Then what happened at that point? or in order for a sequence of, in this case, integrable functions one should have that the limit is different to $\infty$ at every point. thanks for the answers beforehand

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    What kind of limit are you taking?2011-05-16

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In the Lebesgue theory, one generally deals with equivalence classes of functions, rather than with individual functions; maybe a better way of saying this is that when one appears to be referring to a function $f$, one is really referring to the class of functions equivalent to $f$. Functions that differ at just one point are equivalent (indeed, functions that differ at a set of measure zero are equivalent) so it doesn't matter if there's one point $x$ where $f(x)=g(x)=\infty$.

Alternatively, if you insist on dealing with functions, not equivalence classes, then you can't say $f$ is integrable if there's an $x$ with $f(x)=\infty$, because that means $f$ isn't even defined on its putative domain.

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    Okay, I agree, and I was being ambiguous in my comment. Sorry about that (and the many typos). If you're *really* interested, you can find a lengthy discussion in section 24 (especially 241) of volume 2 of [Fremlin's measure theory compendium](http://www.essex.ac.uk/maths/people/fremlin/mt.htm).2011-05-16