Let $(\Omega, F,P) $ be probability space.
If $X\in L^1(\Omega)$ , show that: $\forall \epsilon >0, \exists \sigma >0$ such that: $$ [\forall A\in F : P(A)<\sigma ]\Rightarrow [\int_A |x| dP $$
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To complete the proof you need an additional step to justify why having probability zero will give the result. Hint: it is related to putting the limit inside the integral. – dankmemer Dec 18 '19 at 15:14
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Your statement as it appears seems incomplete. – Sangchul Lee Dec 18 '19 at 16:06
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Take a look at this question – saz Dec 18 '19 at 17:10