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If $f_n\to f$ in $L^1(\mu)\implies f_n $ are uniformly integrable where $\mu $ is positive measure

My Attempt:

Uniformly Integrable family: $\{f_n\}_{n\in A}$ is said to be uniformly integrable if $\forall \epsilon>0$ $\exists \delta >0 $ such that $|\int_E f_n d\mu|$ wherever $\mu (E)<\delta$

let $\epsilon>0$ given

As $f_n\to f$ in $L^1(\mu), \exists N$ that $\int |f_n-f|d\mu<\epsilon/2, \forall n>N$

As for $n<N$

I can find $\delta _i>0$ such that $\mu(E)<\delta_i$

$\int_{E} |f_i|d\mu<\epsilon/2$

FOr $n\geq N$

$\int |f_n|<\int |f-f_n|+\int |f|$

As Form assumption $\int |f-f_n|<\epsilon/2$ for any measurable set

And for f I can always find $\delta_a>0$ such that $\mu(E)<\delta_a$ $\int_E|f|d\mu<\epsilon/2 $

So we can choose $\delta=min \{\delta_1,.....\delta_n,\delta_a\}$

So done

Is I am correct?

I would be thankful if some one help me to find mistake in my proof if there is any

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  • $\begingroup$ this is correct $\endgroup$ – David Bowman Apr 20 at 12:35
  • $\begingroup$ Thanks A lot Sir $\endgroup$ – MathLover Apr 20 at 12:39

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