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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?

  • 1
    Can you apply the property to a set $E \cap \{x| f(x) \geq 0 \}$ and $E \cap \{x| f(x) <0 \}$ separately and combine the results?2012-11-05

1 Answers 1