Saturday, December 10, 2016

real analysis - Prove that limptoinfty|u|p=|u|infty

Let (X,A,μ) be any measure space and let up[1,]Lp(μ). Then limpup=u.


I have already proved the inequality ulim infpup, so I still need to prove lim suppupu.


Following a hint, I have proved that for any sequence qn one has up+qnuqn/(p+qn)up/(p+qn)p.


How can we conclude from this that lim suppupu?

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