Some idea for dealing with this problem? Let (X,Ω,μ)(X,\Omega, \mu) a finite measure space and MM the set of measurable functions over XX. Prove that d(f,g)=∫|f−g|1+|f−g|dμd(f,g)=\int\frac{|f-g|}{1+|f-g|}d\mu

Defines a complete metric over MM. I don’t know how to proceed to show the completeness. Thank you all.

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There are many posts on this problem. Please have a look at them first.

– Jacky Chong

Oct 21 at 2:22

Would you link me to one of them? I’ve been searching for them and I didn’t success. Thanks

– Jan

Oct 21 at 2:31

math.stackexchange.com/questions/753533/…

– Jacky Chong

Oct 21 at 2:33

Proof is essentially the same.

– Jacky Chong

Oct 21 at 2:33

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