If the infinite sum of the norms $||f_n||$ converges then $\sum f_n$ converges to some $f \in \mathcal{L}_2(\Omega)$

Question

I have to prove this theorem:

Let $\{f_n\}_n\subset \mathcal{L}_2(\Omega)$ such that $$\sum_{n=1}^\infty ||f_n|| < +\infty$$

Then, $\sum_{n=1}^\infty f_n$ converges pointwise almost anywhere (i.e. possibly not in a null set) to some function $f \in \mathcal{L}_2(\Omega)$. Moreover, the sum converges to $f$ in $\mathcal{L}_2(\Omega)$. (Note that the first convergence is pointwise and the second is L2 convergence)

I have already shown that $\{S_k^2\}_k$ converges to some Lebesgue integrable function $g$ where $$S_k(x):=\sum_{n=1}^k|f_n(x)|$$ for $x\in\Omega$.

Now I want to deduce that $\sum_{n=1}^\infty f_n$ converges almost everywhere to some $f$ and then prove that $|f|^2$ is Lebesgue integral so then $f\in \mathcal{L}_2(\Omega)$, but I don't know where to start with this step.

Then I would just use dominated convergence theorem for $\{|\sum_{n=1}^k f_n(x)-f(x)|^2\}_{k=1}^\infty$.

Answer 1

Let $S_N = \sum_{n \leq N} |f_n|$. By construction you have that $\{S_N\}$ is a Cauchy sequence in $L^2$; indeed for any $\epsilon > 0$ there is a $N_0(\epsilon)$ sufficiently large such that for $N \geq M \geq N_0(\epsilon)$, you have $$ \|S_N - S_M\|_2 \leq \sum_{N < j \leq M} \|f_n\|_2 \leq \sum_{j > N_0(\epsilon)} \|f_j\|_2 < \epsilon. $$

Since $L^2$ is complete, the sequence $S_N$ converges to a limit $S$ in $L^2$, and consequently there is a subsequence of $S_N$ which converges pointwise almost everywhere to $S$. However, since for such $x$, the sequence $S_N(x)$ is monotone, this implies that except for a set of measure zero, $S_N$ converges pointwise to $S$.

Now note that for those $x$ such that $S_N(x)$ converges, you must have that $\sum_{n \geq 1} f_n(x)$ converges.