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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?

Adam
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  • As Bauer writes, check the proof of Theorem 20.7. – Siminore Jul 20 '14 at 10:55
  • @Siminore Thanks for your suggestion. I already did that and I dont understand how that follows at all from the proof of 20.7, so I didn't mentioned it in my question. Also Bauer indicates that it follows from the second part from the proof - but the proof is not seperated in two parts. – Adam Jul 20 '14 at 11:31

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