Monday, December 26, 2016

analysis - Measure theory limit question





Let (X,M,μ) be a finite positive measure space and f a μ-a.e. strictly positive measurable function on X. If EnM, for n=1,2, and lim, prove that \displaystyle\lim_{n\rightarrow\infty}\mu(E_n)=0.



Answer



Since f is almost everywhere strictly positive, the increasing sequence of sets A_n=\{x\in X:f(x)>1/n\}
has the property that \lim_{n\to\infty} \mu(A_n)=\mu(X). Now $\int_E f ~d\mu0$.



So let \epsilon>0. Choose n so that \mu(X\backslash A_n)<\epsilon/2. For N large enough, \int_{E_N}f~d\mu<\epsilon/(2n) and hence $$\mu(E_N\cap A_n)

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