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Let $(f_n)_{n \in \mathbb N} \subset L^1( X, \Sigma, \mu )$ be a sequence of integrable functions with $\mu $ a finite measure. Suppose further that the sequence is uniformly absolutely continuous that is:

$ \forall \epsilon > 0, \exists \delta_{\epsilon} > 0$ such that for $A \in \Sigma$ with $\mu(A)< \delta$ then $\int_{A}|f_n|d\mu < \epsilon $ for all $n$

Are there any examples of a sequence of functions with the above conditions that are not $L^1$ bounded? that is $$\sup_{n \in \mathbb N}\int |f_n|d\mu = \infty $$

I am having a hard time trying to find such examples. I would really appreciate any hints or suggestions wiht this problem

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If $\mu$ is counting measure on a finite set then uniform absolute continuity trivially holds for any sequence $(f_n)$: take $\delta <1$. Obviously, $(f_n)$ need not be $L^{1}$ bounded.

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  • $\begingroup$ I didn´t think about the counting measure! Nice example! This leaves me thinking if we can find such example in $[0,1]$ with the Lebesgue measure $\endgroup$ – user128422 Jul 9 at 15:37

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