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.

Show that if $f : E \rightarrow [0,\infty]$, $\lim \limits _{k\rightarrow \infty} f_k = f$ on $E$, and $f_k \leq f$ on $E$ for each $k \in N$, then $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k = \int \limits _E f$

An idea was to show that $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \leq \int \limits _E f$ and $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \geq \int \limits _E f$. I am able to prove $\lim \limits _{k\rightarrow \infty} \int \limits _E f_k \leq \int \limits _E f$ but am struggling to prove the second condition. My idea was to use fatou's lemma to get $$ \int \limits _E \liminf \limits _{k\rightarrow \infty} f_k = \int \limits _E \lim \limits _{k\rightarrow \infty} f_k = \int \limits _E f \leq \liminf \limits _{k\rightarrow \infty} \int \limits _E f_k = \lim \limits _{k\rightarrow \infty} \int \limits _E f_k $$ but I don't know how to show $\liminf \limits _{k\rightarrow \infty} f_k = \lim \limits _{k\rightarrow \infty} f_k$, besides showing $\liminf \limits _{k\rightarrow \infty} f_k = \limsup \limits _{k\rightarrow \infty} f_k$ .

Any ideas on how I could finish this? Also, is my approach wrong? Could I do it a better way?

share|improve this question
    
Provided that $\displaystyle \lim_{k \to \infty} f_k$ exists, it always equals $\displaystyle \liminf_{k \to \infty} f_k$. Same for real numbers in place of $f_k$. –  Lord_Farin Oct 16 '12 at 18:11
    
Doesn't $\lim \limits _{k\rightarrow \infty} f_k = f$ imply that the limit exists? –  rioneye Oct 17 '12 at 21:00
    
It does, so you may infer the sought equality directly. You may want to read up on properties of $\lim, \liminf, \limsup$ and how they interact. –  Lord_Farin Oct 17 '12 at 21:56
add comment

1 Answer

up vote 1 down vote accepted

Writing $\lim_{k\to\infty}\int_Ef_k$ is not correct once we have not proved that the limit exists. Anyway, we can apply Fatou lemma to $f-f_k$ to get an inequality involving $\limsup_{k\to +\infty}\int_Ef_k$. With what we get: $\limsup_{k\to+\infty}\int_E f_k\leq \int_E f\leq \liminf_{k\to+\infty}\int_E f_k$. (this works when the integral of $f$ is finite). If $f$ is not integrable then the first application of Fatou lemma shows that $\int_Ef_k$ converges to $+\infty$.

share|improve this answer
    
Could you elaborate, since I don't quite understand how what you said will help to solve the problem. I apologize if it is supposed to be obvious, proofs are my bane. –  rioneye Oct 16 '12 at 19:43
    
I didn't give a lot of details as it's homework. Now I edited. –  Davide Giraudo Oct 16 '12 at 19:49
add comment

Your Answer

 
discard

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.