Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
What kind of convergence are you assuming if you write $f_n\rightarrow f$? –  user20266 May 27 '12 at 8:34
@Thomas Pointwise. I forgot to mention this. –  Polymorpher May 27 '12 at 8:35
here I give you a example X=(0,1),$f_{n}=n^{2}\chi_{(0,1/n)}$, the claim in general is wrong –  yaoxiao May 27 '12 at 8:38
@Polymorpher this will be true, just consider Fatou theorem, you will get what you want –  yaoxiao May 27 '12 at 8:45
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). –  Najib Idrissi May 27 '12 at 8:47
show 5 more comments

1 Answer 1

up vote 2 down vote accepted

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 $g+g_n-|f-f_n|$.

share|improve this answer
It does not meet the condition $\int{g_{n}}\to\int{g}$ –  Polymorpher May 27 '12 at 9:45
What is $g$ in this context? –  Davide Giraudo May 27 '12 at 9:45
It is the limit of $g_{n}$. Sorry, I forgot to mention that... –  Polymorpher May 27 '12 at 9:47
Then show $\limsup$ and $\liminf$ equals, from inequalities in both directions. –  Polymorpher May 27 '12 at 10:03
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. –  Sam May 27 '12 at 12:06
show 6 more comments

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.