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 fn does not converge uniformly to f on the set [0,∞), a set of infinite measure. And so we conclude that fn does not converge almost uniformly to f.
If we choose ϵ=0 and η=5 then for all N(ϵ,η)∈N we have that for n≥N.
μ(x∈D:|fn|≥ϵ)=μ(R)=+∞≥5.
So we see that fn 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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