How can I show that if family of $f$ is uniformly integrable then so is {$|f|$}?
$($by uniformly integrablity: $\forall \epsilon>0 \ \exists \delta>0: |\int_Ef|<\epsilon,\mu(E)<\delta)$
My attempt:
$|\int_E f|=|\int_{E^+} f^+-\int_{E^-} f^-| \leq \epsilon$
$|\int_{E^+} f^++\int_{E^-} f^--2\int_{E^-} f^-| \leq \epsilon$
$|\int_{E} |f|-2\int_{E^-} f^-| \leq \epsilon$
$\mu(E^-)\leq \mu(E)\leq \delta$ so I want to conclude $2\int_{E^-} f^-$ is negligible, thus $|\int_{E} |f|| \leq \epsilon$ and get done. Is it OK?