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Show that in $L_1(\mathbb{R})$ every absolutely convergent series converges.

My try:

Take some absolutely convergent series in $L_1(\mathbb{R})$. Then we have: $$ \sum_{n=0}^\infty \Vert f_n\Vert = \sum_{n=0}^\infty \int_{\mathbb{R}} |f_n|\ \mathrm{d} \lambda = \int \sum_{n=0}^\infty |f_n| \ \mathrm{d} \lambda <\infty.$$ This can be true only if $\sum_{n=0}^\infty |f_n (x)| <\infty $ for $\lambda$-almost all $x$. As $\sum_{n=0}^\infty |f_n (x)| $ is a monotone series in $\mathbb{R}$, it now has to be convergent.

For a series in $\mathbb{R}$ we know that absolute convergence implies convergence. So now we can define $F(x):= \sum_{n=0}^\infty f_n(x)$ for $\lambda$-almost all $x$, and assign to $F$ some arbitrary value for all the other $x$. Then $\sum_{n=0}^m f_n\to F$ pointwise for $\lambda$-almost all $x$, which means that $$\Vert \sum_{n=0}^m f_n-F\Vert = \int_{\mathbb{R}} |\sum_{n=0}^m f_n-F|\ \mathrm{d}\lambda \to 0,$$ so $\sum_{n=0}^\infty f_n$ is a convergent series.

Is this proof correct? I'm not sure about reversing integration and summation at the beginning. Any help would be appreciated.

user161518
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1 Answers1

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The proof is mostly okay, until the end.

You say "I'm not sure about reversing integration and summation at the beginning" - whenever you say this, that means you have to prove it more carefully! The linearity of the integral only lets you interchange limits with finite sums, so you can't just rely on that. In this case you can use the monotone convergence theorem: if you let $S_m = \sum_{n=0}^m |f_n|$, then $\{S_m\}$ is a monotone increasing sequence of nonnegative measurable functions, converging pointwise to $\sum_{n=0}^\infty |f_n|$.

At the end there is a big gap, where you have the words "which means that". Showing that $\sum_{n=0}^m f_n \to F$ pointwise is definitely not enough by itself to imply that $\int \left|\sum_{n=0}^m f_n - F\right| \to 0$. Surely you know lots of sequences which converge pointwise but their integrals don't converge to the integral of the pointwise limit. There is a fair amount of work still to do here.

Nate Eldredge
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  • To fill the big gap, could we maybe apply Dominated Convergence, as $|\sum_{n=0}^m f_n|\leq \sum_{n=0}^m |f_n|\leq \sum_{n=0}^\infty |f_n|$? Then I would still need to prove that $\sum_{n=0}^\infty |f_n|$ is integrable, but this follows from the Monotone Convergence statement I need to prove the integration/summation-reversal at the beginnning, right? – user161518 Oct 23 '15 at 14:36
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    @user161518: Yes, something like that should work. – Nate Eldredge Oct 23 '15 at 14:39
  • Great! Then concerning this Monotone Convergence statement. Can we in fact say that $\sum_{n=0}^m |f_n|\to \sum_{n=0}^\infty |f_n|$? Because we don't yet know that $\sum_{n=0}^\infty |f_n(x)|$ is finite for all $x$ right? Or doesn't it have to be finite at all? – user161518 Oct 23 '15 at 14:43
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    @user161518: If you read the MCT carefully, it doesn't require the limit function to be finite. – Nate Eldredge Oct 23 '15 at 14:47
  • Okay, so you can apply MCT to every monotone sequence of measurable functions? – user161518 Oct 23 '15 at 14:50
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    @user161518: They also have to be nonnegative. – Nate Eldredge Oct 23 '15 at 14:52