Strengthen a condition (something like uniform integrability)

Question

Suppose $\{f_k\}$ is a sequence of integrable functions on $(0,1)$ such that given $\epsilon>0$, there exists $\delta>0$ such that if $|E|<\delta$, then $$\left|\int_E f_k\right|<\epsilon$$ for all $k$.

Is it possible to deduce that:

given $\epsilon>0$, there exists $\delta>0$ such that if $|E|<\delta$, then $$\int_E |f_k|<\epsilon$$ for all $k$?

Note that this is stronger than the previous statement since $\left|\int_E f_k\right|\leq\int_E|f_k|$.

Thanks!

Answer 1

Yes, (a part from a factor of 2 on $\epsilon$). Let $\epsilon>0$ and find $\delta>0$ so that the first condition is satisfied.

For given $k$ let $I_+=\{f_k\geq 0\}$ and $I_-=\{f_k<0\}$. Then for any $E$ with $|E|<\delta$ $$ \int_E |f_k| = \int_{E\cap I_+} f_k - \int_{E\cap I_-} f_k \leq 2 \epsilon$$.