I found in the measure and integration theory book from Bauer (Remark 6, §20) that if $\mu$ is a finite measure, then the weak convergence of a sequence $(f_n)$ is equivalent to $$ \lim_{n,m \to \infty} \mu(\{ |f_m-f_n| \geq \alpha \})=0 $$ for all $\alpha >0$.
It is clear to me that the condition is necessary because of $$ \{ |f_m -f_n | \geq \alpha \} \subset \{ |f_m -f| \geq \alpha/2 \} \cup \{ |f_n -f| \geq \alpha/2 \}.$$ But how can I see that this is also sufficient?