# Difficult example of Uniformly absolute continuous sequence but not $L^1$ bounded

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

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.
• 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 – user128422 Jul 9 at 15:37