Monday, 1 May 2017

real analysis - fn is uniformly integrable if and only if supnint|fn|,dmu<infty and fn is uniformly absolutely continuous?

Let (X,A,μ) be a measure space. A family of measurable functions {fn} is uniformly integrable if given ϵ there exists M such that{x:|fn(x)|>M}|fn(x)|dμ<ϵ

for each n. The sequence is uniformly absolutely continuous if given ϵ there exists δ such that|Afndμ|<ϵ
for each n if μ(A)<δ.



Suppose μ is a finite measure. How do I see that {fn} is uniformly integrable if and only if supn|fn|dμ< and {fn} is uniformly absolutely continuous?

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