Problem. Let (X,μ) be a probability measure space such that fnμ→0 and that Var(fn)=1, prove that E(fn)→0.
Here Var(fn)=∫Xf2ndμ−(∫Xfndμ)2 and E(fn)=∫Xfndμ.
In the sequel we denote ∫f=∫Xfdμ and ∫Af=∫Afdμ. Then E(fn) is also written as ∫fn.
In the proof below we need an elementary inequality from probability that easily follows from Chebyshev's inequality: μ{x∈X:|fn(x)−E(fn)|≥a}≤Var(fn)a2.
Proof.
Suppose on the contrary that E(fn)↛0, then there is an ϵ0>0 such that |∫fnk|≥ϵ0. It follows that ϵ0≤|∫{|fnk|≥1}fnk|+|∫{|fnk|<1}fnk|≤√∫{|fnk|≥1}f2nk⋅√μ{|fnk|≥1}+∫|fnk|χ{|fnk|<1}≤√∫f2nk⋅√μ{|fnk|≥1}+∫|fnk|χ{|fnk|<1}. Since fnk→0 in measure, there is a subsequence fnkp that converges to 0 pointwise μ-a.e.. By Lebesgue Dominated Convergence Theorem, ∫|fnkp|χ{|fnkp|<1}→0. So the above becomes ϵ0≤¯limp→∞√∫f2nkp⋅√μ{|fnkp|≥1}. We will finish the proof by showing ∫fn is in fact bounded in n, and so is ∫f2n since 1=Var(fn)=∫f2n−(∫fn)2, and then the above inequality yields a contradiction that ϵ0≤0.
Now we turn to the boundedness of {∫fn}, the only tricky part of this problem. For this, observe that for any measurable f,g on X, μ{|f|≥2ϵ}≤μ{|f−g|+|g|≥2ϵ}≤μ{|f−g|≥ϵ}+μ{|g|≥ϵ}. In particular, let's take f to be the constant function E(fn), and g=fn, then μ{x∈X:|E(fn)|≥2ϵ}≤μ{|E(fn)−fn|≥ϵ}+μ{|fn|≥ϵ}≤1ϵ2+μ{|fn|≥ϵ}, the last inequality follows from μ{|fn−E(fn)|≥ϵ}≤Var(fn)ϵ2. Taking ¯lim on both sides above, we get ¯limμ{x∈X:|E(fn)|≥2ϵ}≤1ϵ2 by convergence in measure. Take ϵ=2, there is an N such that for every n>N, μ{x∈X:|E(fn)|≥2⋅2}<1, and therefore {x∈X:|E(fn)|≥4}=∅, and thus |E(fn)|<4 for n>N.◼
Remark. After examining the proof the condition that Var(fn)=1 can be replaced by that the sequence {Var(fn)}∞n=1 is bounded, since we can generalize (???) to
¯limμ{x∈X:|E(fn)|≥2ϵ}≤Var(fn)ϵ2.
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