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If $X_n$ are independent random variables, then does $\sum_n \mathbb{E}(X_n)=\mathbb{E}(\sum_n X_n)$?

This is not a homework problem but rather a question I had. If it is not true, what are the weakest conditions that would make it true? I am aware that if either are absolutely summable, then fubini tells us the answer is yes. I am more interested in the case that fubini does not hold. What weaker conditions can we impose but use the independence of the random variables?

anonymous
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    Indeed, your proposition holds regardless of the dependence of the random variables. For a more general and complete argument you may wanna see the question tagged above. – Nighty Oct 21 '14 at 07:29
  • I am more interested when the conditions of fubini aren't satisfied. Can the independence give us new weaker conditions? – anonymous Oct 21 '14 at 08:46
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    @DavideGiraudo Why did you see fit to reopen this obvious duplicate? – Did Oct 21 '14 at 09:51
  • @Did I should have red more carefully the revision: it is still a duplicate after this revision. – Davide Giraudo Oct 21 '14 at 14:35

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