If $\sum \|f_n -f \|_{L^1} < \infty$ then $f_n \rightarrow f$ almost uniformly

Question

Consider $(X,m)$ a measure space, $f_n, f : X \rightarrow \mathbb R$ s.t. $\sum_{n=1}^{\infty} \|f_n -f \|_{L^1} < \infty.$ How to show that $f_n \rightarrow f$ almost uniformly?

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I will have proved the result if for any $\varepsilon > 0$ I can find $A \subseteq X$ s.t. $m(A) < \varepsilon$ and $$\sup_{x \in A^c} f_n(x) \xrightarrow n f.$$ How should I build $A$?

Answer 1

Hints: Fix $\epsilon>0$.

1. Fix $k \in \mathbb{N}$. Using Markov's inequality, show that $$A_N^k := \left\{x; \exists n \geq N: |f_n(x)-f(x)| \geq \frac{1}{k} \right\}$$ satisfies $$m(A_N^k) \leq k \sum_{n=N}^{\infty} \|f_n-f\|_{L^1}.$$
2. Conclude from the first step that there exists $N=N(k)$ such that $m(A_{N(k)}^k) \leq \epsilon 2^{-k}$.
3. Set $$A := \bigcup_{k \in \mathbb{N}} A_{N(k)}^k.$$ Show that $$m(A) \leq \epsilon$$ and that $f_n \to f$ uniformly on $X \backslash A$.