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Let $f$ be continuous on $[a,b]$ and has finite derivative a.e. on $[a,b]$. Let $f_n(x)=n[f(x+1/n)-f(x)] $ s.t. $f_n$ be uniformly integrable on $[a,b]$.

I want to show : $f'$ is Lebesgue integrable.

(I noted that $f_n\rightarrow f'$ pointwise.)

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    A pointwise limit of a uniformly integrable sequence is *always* integrable.2012-12-10

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