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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?

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  • $\begingroup$ 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 $f_k$. $\endgroup$ – Lord_Farin Oct 16 '12 at 18:11
  • $\begingroup$ Doesn't $\lim \limits _{k\rightarrow \infty} f_k = f$ imply that the limit exists? $\endgroup$ – rioneye Oct 17 '12 at 21:00
  • $\begingroup$ 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. $\endgroup$ – Lord_Farin Oct 17 '12 at 21:56
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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 $+\infty$.

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  • $\begingroup$ 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. $\endgroup$ – rioneye Oct 16 '12 at 19:43
  • $\begingroup$ I didn't give a lot of details as it's homework. Now I edited. $\endgroup$ – Davide Giraudo Oct 16 '12 at 19:49

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