Consider a sequence (fn) defined on R by fn=χ[n,n+1], n∈N and the function f≡0. Does fn converge to f almost everywhere, almost uniformly or in measure?
fn→f almost everywhere is the same as saying that fn→f pointwise almost everywhere, i.e. on a subset whose complement has measure zero. Given ϵ>0 and x∈R we observe that if x≥0 then for some n0∈N we have that $n_0\leq x
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:
No comments:
Post a Comment