# limit of Lebesgue integrable functions

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? –  Qiaochu Yuan May 16 '11 at 6:18

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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I disagree with your second paragraph. It makes perfect sense to look at extended real-valued functions (at the cost of dealing with a vector space before identifying functions if they agree a.e.). Since the integral is defined as a limit of simple functions, we can define the integral even if $f$ takes on the values $\pm \infty$ (as long as the set where these values are attained are negligible) and $f$ agrees a.e. with an integrable function. –  t.b. May 16 '11 at 6:30
If you let your functions take on infinite values, why not let your integrals take on infinite values, too? Why not say the integral of the function that's constantly infinity is infinity? –  Gerry Myerson May 16 '11 at 7:58
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. –  t.b. May 16 '11 at 8:28