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.)
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.)