Wednesday, 26 July 2017

How to show that the given sequence of functions converges to f almost everywhere, almost uniformly and in measure?

Consider a sequence (fn) defined on R by fn=χ[n,n+1], nN and the function f0. Does fn converge to f almost everywhere, almost uniformly or in measure?



fnf almost everywhere is the same as saying that fnf pointwise almost everywhere, i.e. on a subset whose complement has measure zero. Given ϵ>0 and xR we observe that if x0 then for some n0N we have that $n_0\leq xOn the other hand if x<0 then fn(x)=0 and so we see that lim for all x\in \mathbb{R}. Since the complement of \mathbb{R} is \emptyset which has measure 0 we conclude that f_n converges to f almost everywhere.




Now f_n does not converge uniformly to f on the set [0,\infty), a set of infinite measure. And so we conclude that f_n does not converge almost uniformly to f.



If we choose \epsilon = 0 and \eta =5 then for all N(\epsilon,\eta)\in \mathbb{N} we have that for n\geq N.
\mu(x\in D:|f_n|\geq \epsilon)=\mu(\mathbb{R})=+\infty\geq 5.



So we see that f_n does not converge to f in measure.



Is this solution correct?




This is problem 39 page 48 on the following pdf https://huynhcam.files.wordpress.com/2013/07/anhquangle-measure-and-integration-full-www-mathvn-com.pdf



The definition for convergence in measure used there is:



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