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I am trying to show given $f_{n}\to f$ pointwise a.e. and $\int{f}<\infty$, it follows the sequence {$\int{f_{n}}$} has a limit. But I am not sure if extra condition is required. Can anyone give me a counter-example, or a simple proof?

Edit: a counter-example is already found. What if there is an extra condition $|f_{n}|\le g_{n}$ and $\int{g_{n}}\to \int{g} \le \infty$ ?

Edit2: It is already shown that this can be proven by Fatou's lemma. Thanks everyone who helped me.

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    What kind of convergence are you assuming if you write $f_n\rightarrow f$?2012-05-27
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    @Thomas Pointwise. I forgot to mention this.2012-05-27
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    here I give you a example X=(0,1),$f_{n}=n^{2}\chi_{(0,1/n)}$, the claim in general is wrong2012-05-27
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    You might be interested in this: https://en.wikipedia.org/wiki/Dominated_convergence2012-05-27
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    @yaoxiao What if I put an extra condition: $|f_{n}|\le g_{n}$ and $\int{g_{n}}\to \int{g} \le \infty$2012-05-27
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    @Polymorpher this will be true, just consider Fatou theorem, you will get what you want2012-05-27
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    @N.I We do not have a single function to bound each $f_{n}$ therefore this theorem does not apply.2012-05-27
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    My point is, you need the extra condition. Since someone already gave a counter-example, I gave you one possible extra condition (and also I missed you edit by 6 minutes).2012-05-27
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    @yaoxiao I know it is very easy to use Fatou's lemma to show $\int{f_{n}}=\int{f}$. I was trying to avoid that. But it looks like this cannot be avoided. Thanks!2012-05-27
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    [related](http://math.stackexchange.com/questions/108313/an-application-of-the-general-lebesgue-dominated-convergence-theorem)2012-05-28

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After precision have been brought here is a proof. We have $g_n-f_n\geq 0$ hence by Fatou lemma $$\int \liminf_n (g_n-f_n)\leq \liminf_n\int (g_n-f_n) $$ so $\int g-\int f\leq \int g+\liminf_n \int (-f_n)$ and $\limsup_n \int f_n\leq \int f$.

Since $g_n+f_n\geq 0$, we apply the previous job to $-f_n$ to get $-\liminf_n \int f_n\leq -\int f$ hence $\lim_{n\to +\infty}\int f_n=\int f$.

A shorter way suggested by @Sam L. is to apply Fatou lemma to $g+g_n-|f-f_n|$.

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    It does not meet the condition $\int{g_{n}}\to\int{g}$2012-05-27
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    What is $g$ in this context?2012-05-27
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    It is the limit of $g_{n}$. Sorry, I forgot to mention that...2012-05-27
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    Pointwise limit?2012-05-27
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    Yes, a.e pointwise limit. I worked out a proof using Fatou's lemma to show this limit exists. Thank you for your help! I have edited my question.2012-05-27
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    To apply Fatou's lemma you need $f$ non-negative. Is it the case?2012-05-27
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    It can be applied on $g-f$2012-05-27
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    But it won't prove that $\int f_n$ is convergent.2012-05-27
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    Then show $\limsup$ and $\liminf$ equals, from inequalities in both directions.2012-05-27
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    I think it would be easier to consider $g+g_n - |f - f_n|\ge 0$ and apply Fatou to this. Then you'd get convergence $f_n\to f$ in $L^1$ in one go.2012-05-27
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    Indeed, thanks to that we just apply Fatou one time.2012-05-27