I'm learning from Stein's Real analysis, Chapter 2. When reading the Riesz-Fischer (pp70), I found part of the proof confusing:
Suppose$\{f_n\}$ is a sequence of integrable functions，and is Cauchy
in $L_1$ space$\to \forall \epsilon > 0, \exists N \in \mathbb{N}^{+}$ s.t.
$\forall n, m > N(\epsilon), ||f_n – f_m|| < \epsilon$ .Then choose a subseq ${f_{n_k}}$ s.t. $||f_{n_{k + 1}} – f_{n_k}||
\leq 2^{-k}, \forall k \geq 1$.Define $f := f_{n_1}(x) + \sum_{k = 1}^\infty (f_{n_{k+1}}(x)
-f_{n_{k}}(x)) $,$g := |f_{n_1}(x)| + \sum_{k = 1}^\infty (|f_{n_{k+1}}(x) -f_{n_{k}}(x)|)$
Easy to see that $\int_{\mathbb{R}}g < \infty$, so $g$ is integrable。Since
$|f| \leq g$, $f$ is integrable as well.Also, the series defining $f$ converges almost everywhere…
What does it mean by the series defining $f$ converges almost everywhere? How can we see that?
Best Answer
Since $\int_{\mathbb{R}}g < \infty$, $g$ is finite almost everywhere.
It means that $|f_{n_1}(x)| + \sum_{k = 1}^\infty (|f_{n_{k+1}}(x) -f_{n_{k}}(x)|)$ converges almost everywhere.
So, $f_{n_1}(x) + \sum_{k = 1}^\infty (f_{n_{k+1}}(x) -f_{n_{k}}(x)) $ converges (absolutely) almost everywhere.